ON SOME NONLOCAL EVOLUTION EQUATIONS ARISING IN MATERIALS SCIENCE

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1 ON SOME NONLOCAL EVOLUTION EQUATIONS ARISING IN MATERIALS SCIENCE PETER W. BATES Abstract. Equations for a material that can exist stably in one of two homogeneous states are derived from a microscopic or lattice viewpoint with the assumption that the evolution follows a gradient flow of the free energy with respect to some metric. Alternatively, Newtonian dynamics can be considered. The resulting lattice dynamical systems are analyzed, as are equations on the continuum where the lattice interaction energy is viewed as an approximation to a Riemann integral. These equations are lattice or nonlocal versions of the Allen-Cahn, Cahn-Hilliard, Phase- Field, or Klein-Gordon equations. Some results presented here provide for the well-posedness of the equations, while others give asymptotics or quantitative behavior of special solutions, such as traveling waves or pulses. This summarizes results previously reported in papers with coauthors Xinfu Chen, Adam Chmaj, Jianlong Han, Chunlei Zhang, and Guangyu Zhao. 1. Introduction We view a material sample as a collection of atoms occupying an n- dimensional lattice Λ. Figure 1. Lattice with long-range interactions 1 These atoms will be assigned spin A or B but we view this as an order parameter that could represent many different things, such as true spin, local concentration or degree of solidification, etc. (when each atom is really a small block of material itself). Allowing for fluctuations, we take

2 2 PETER W. BATES We shall obtain a complete solution of the problem... if we can express the free energy at each point as a function of the density at that point and of the differences of density in the neighboring phases, out to a distance limited by the range over which the molecular forces act. It is possible that some molecular forces act, albeit with very small strength, at great distances and we adopt that point of view, choosing to include all pairwise interactions. The following reasoning was described in more detail in [11] and [12] but we include a brief description here for completeness. The Helmholtz free energy of a state is given by E = H T S, where H = interaction energy, T = absolute temperature, and S = total entropy. We include all pairwise interaction, allowing for the possibility that pairs of type A interact differently than pairs of type B and both differently from the interaction of mixed pairs. Thus, H(a) 1 [ J AA (r r )a(r)a(r ) + J BB (r r )(1 a(r))(1 a(r ))+ 2 r,r Λ ] J AB (r r )(a(r)(1 a(r )) + a(r )(1 a(r))). We expect the interaction, through the J s, to be symmetric and translationinvariant, but possibly anisotropic. Rearranging: H = 1 J(r r )(a(r) a(r )) 2 4 r,r Λ D 1 2 r Λ(a(r) 2 a(r)) + d a(r) + const. r Λ where J(r) = J AA (r) + J BB (r) 2J AB (r), D = J(r), and d = (J BB (r) J AA (r))/2. At site r the entropy s(a(r)) for an particles in N identical sites is given by e Ns/K N! = (an)!(n an)! where K is Boltzman s constant. Hence, s(a) K[a ln a + (1 a) ln(1 a)]. The total entropy, S(a) = r Λ s(a(r)) and so E(a) = H T S = 1 J(r r )(a(r) a(r )) r,r Λ [ KT {a(r) ln a(r) + (1 a(r)) ln(1 a(r))} D(a(r) 2 a(r)) + da(r) ]. r Λ

3 There is a critical temperature T c such that for T T c the term [ ] is strictly convex and so there is a unique homogeneous state which minimizes E(a), while for T < T c, this term has two local minima and so two distinct a-states (say α < β) give spatially homogeneous local minimizers of E. This is the origin of phase transition in spin systems (e.g. ferromagnets.) 3 Figure 2 We will fix T < T c. If we were to take continuum limit by using a scaling so that the summation could be viewed as an approximation to a Riemann integral, then we would obtain a free energy in the isothermal case of the form E(u) = 1 J(x y)(u(x) u(y)) 2 dxdy + F (u)dx, 4 where F is a double well function, having minima at ±1 (after changing variables), and J is assumed to be integrable with positive integral and with J( x) = J(x). It is interesting to compare with Ginzburg-Landau functional: ( ε2 2 u 2 + F (u))dx. This is easily obtained from the above nonlocal energy by assuming the atomic interaction is short ranged so that for each state u, one could be justified by approximating (u(x) u(y)) (x y) u(x). With that, the coefficient ε 2 in the energy is a second moment of J in the isotropic case. In fact van der Waal s took this approach. The resulting Euler-Lagrange equation, ε 2 u F (u) =, has been well studied and provides some insight into phase transitions. We do not make this short-range approximation however, believing that while it may be good for a single smooth function u, it is not a good approximation in the operator sense. It is worth noting here that for several results we do not require that J be nonnegative, although it is assumed to have positive integral (or sum) and we sometimes will assume that it has a positive second moment. Away from equilibrium we take as a fundamental principle the postulate that a material structure evolves in such a way that its free energy decreases as quickly as possible. That is, the spatial function u will evolve in such a way that E(u) decreases, and does so optimally in some sense as u evolves

4 4 PETER W. BATES in a function space, X. This suggests the evolution law (1.1) u t = grade(u), where grad E(u) X, the dual of X, is defined by < grad E(u), v >= d dh E(u + hv) h=. If X = L 2 then (1.1) becomes what we call the nonlocal Allen-Cahn equation, u (1.2) t = J u Du F (u), where * is convolution and D = J is assumed positive. The above equation is for the case when the domain = R n but for general we have u (1.3) t = J u u J(x y)dy F (u), where J u(x) J(x y)u(y)dy. If we had made the Ginzburg-Landau approximation, the resulting gradient flow would be the Allen-Cahn equation [5], u t = ε2 u F (u). Note that the operator J u u J(x y)dy may be thought of as an approximation to the Laplacian, especially in the case J, since then it is a nonpositive selfadjoint operator which has a maximum principle. However, unlike the Laplacian, it is bounded and so (1.3) does not smooth in forward time and has solutions that exist locally backwards in time. If we retain the infinite lattice model instead of moving to Riemann integrals, the equation is similar but in this discrete case convolution is given by J u(r) = s Λ J(r s)u(s). For both the continuous and lattice versions, there is now a large body of work giving qualitative behavior of solutions, traveling waves, propagation failure, stability and pattern formation (see, e.g., [59], [11], [12], [15], [27], [35], [36], [37], [28], [3], [29], [1], [9], and the references therein). There is other recent work on nonlocal equations (see [71], [72], and [5]) but the earliest is perhaps that by Weinberger [78]. In the case that u represents local concentration of one species in a binary alloy, then, with the idea of conserving species, we take X = H 1 (the dual of H 1 with zero mean). Then (1.1) becomes what we call the nonlocal Cahn-Hilliard equation u (1.4) t = (J u u J(x y)dy F (u)).

5 Of course, the original Cahn-Hilliard equations, introduced in [24], has undergone intensive study (see [39], [4], [69], [68], [75], [26], [2], [3], [13], [14], [16], and the references therein) but little has been written on the nonlocal version. To the best of our knowledge, the first was Giacomin and Liebowitz [53], [54], but more recently other results have appeared (see [52] and [32]). Here, we extend some of those results, summarizing the findings in [17] and [18] on the well posedness of (1.4) and long term behavior of solutions. When temperature is allowed to evolve and latent heat of fusion is included in the model then the free energy, E, is often taken to be 1 (1.5) J(x y)(u(x) u(y)) 2 dxdy + (F (u(x)) θ2 )dx, where u represents degree of solidification, θ is absolute temperature, and l is a latent heat coefficient. The internal energy density is given by e = θ + lu and in order to conserve the total internal energy, I e, the simplest gradient flow is with respect to (u, e) L 2 H 1. This leads to the nonlocal phase field system: (1.6) (1.7) u t = J u u J(x y)dy F (u) + lθ, (θ + lu) t = θ. The local phase-field system, introduced by Fix [47], Langer [66], and Caginalp [22], has also undergone much analysis and generalization (see, e.g., Caginalp and Fife [23], Penrose and Fife [7], Kenmochi and Kubo [6], Colli and Laurencot [33], Colli and Sprekels, [34], etc.) and still is finding many new and important applications. With hysteresis and nonlocal effects there is also the work of Krejči, Sprekels, and S. Zheng, [61], [62], [74] and some previous results in [8], all of which influenced this work on well-posedness and long term behavior of solutions for the nonlocal version (1.6), (1.7). We would also like to point out recent interesting results in [45] giving stabilization in the case of analytic nonlinearity. Finally, we are interested in Newtonian dynamics with a force derived from the isothermal energy according to 2 u t 2 = grade(u). In L 2 this leads to a nonlocal wave equation 2 u (1.8) t 2 = J u u J(x y)dy F (u). For this equation we will establish the existence of traveling pulses with = R and J replaced by a large amplitude, short range kernel, namely, 1 j ε 2 ε where j ε (x) = 1 ε J( x ε ). Thus, we will consider 2 u (1.9) t 2 = 1 ε 2 (j ε u u) F (u). 5

6 6 PETER W. BATES In the discrete case, the corresponding equation takes the form (1.1) ü n = 1 ε 2 k= α k u n k F (u n ), n Z where ε > and the coefficients α k satisfy k= α k =, α <, α k = α k, α k k 2 = d >. This may be viewed as a generalized lattice Klein- k 1 Gordon equation. While several studies exist for versions of lattice Klein- Gordon equations (e.g., see [25], [44], [49], [63], [73], and [76], etc.), to the best of our knowledge, there are no prior results for (1.9). In the following three sections we outline results for the nonlocal Cahn- Hilliard, phase-field, and Klein-Gordon equations, respectively. In Section 4. we also prove existence of traveling pulses for the Klein Gordon lattice system. Since the results can seem disembodied without an idea of what lies behind, we give some of the details of the proofs, and where details are lacking, we indicate the route. Our hope is that the reader will gain an appreciation for the variety of techniques that may be brought to bear on these nonlocal evolution equations. Missing here are variational methods but the reader may turn to [1], for instance, to see that those methods may also be applied in some cases. 2. Nonlocal Cahn-Hilliard Equation The first issue to address is whether or not (1.4) is well-posed with suitable boundary conditions. These results are to be found in more detail in papers with Jianlog Han, [17] and [18]. Since the equation is second order in space (while the usual Cahn-Hilliard equation is fourth order), only one boundary condition is expected to be necessary and sufficient for existence and uniqueness of the solution. We will therefore consider both the Dirichlet and no-flux boundary condition, the latter being more natural in the sense that species should then be conserved. Thus, we consider (2.1) with either (2.2) or u t = (J u u u = on J(x y)dy f(u)) (2.3) n ( J(x y)dyu(x) J(x y)u(y)dy + f(u)) = on, where f = F is of bistable type (e.g., f(u) = u u 3 ). This second condition of Neumann type (2.3) may look peculiar but simply states that the chemical

7 potential has no flux across the boundary. We append the initial condition u(x, ) = u (x), for x. We treat the Neumann problem first, discussing the main points of [17]. In order to prove the existence of a classical solution to (2.1) (2.3) we need the initial data to satisfy the boundary condition. So we assume u (x) C 2+β, 2+β 2 ( ) for some β >, and u (x) satisfies the compatibility condition: ( J(x y)dyu (x) J(x y)u (y)dy + f(u )) (2.4) = on. n Rewrite the initial-boundary value problem as u t = a (x, u) u + b(x, u, u) in, t >, a (x, u) u n + a(x) n u(x) J(x y) (2.5) n u(y)dy = on, t >, where u(x, ) = u (x), 7 a(x, u) = a(x) + f (u), a(x) = J(x y)dy, b(x, u, u) = 2 a u + f (u) u 2 + u a ( J) u. We assume the following conditions: (A 1 ) a(x) C 2+β ( ), f C 2+β (R). (A 2 ) There exist c 1 >, c 2 >, and r > such that a(x, u) = a(x) + f (u) c 1 + c 2 u 2r. (A 3 ) is of class C 2+β. With regard to (A 2 ), note that if a(x) + f (u(x, t)) <, for some (x, t) then there is no solution beyond that point in general, since the equation is essentially a backward heat equation. Note also that (A 2 ) implies (2.6) F (u) = u f(s)ds c 3 u 2r+2 c 4 for some positive constants c 3, and c 4. For any T >, denote Q T = (, T ). We first establish an a priori bound for solutions of (2.1) (2.3). Theorem 2.1. If u(x, t) C 2,1 ( Q T ) is a solution of equation (2.1)-(2.3), then (2.7) max Q T u(x, t) C(u )

8 8 PETER W. BATES for some constant C(u ). In order to prove the theorem, we need the following lemma. Lemma 2.2. If u(x, t) C 2,1 ( Q T ) is a solution of equation (2.1) (2.3), then there is a constant C(u ) such that (2.8) sup u(, t) q C(u ) t T for any q 2r + 2. Proof. Let (2.9) E(u) = 1 4 J(x y)(u(x) u(y)) 2 dxdy + F (u(x))dx. Since we have a gradient flow Therefore E(u) E(u ), i.e., de(u) dt J(x y)(u(x) u(y)) 2 dxdy + F (u(x))dx J(x y)(u (x) u (y)) 2 dxdy + F (u (x))dx. From condition (A 1 ), (2.6), and Young s inequality, we obtain u 2r+2 dx C(u ). Since this is true for any t >, we have sup u 2r+2 dx C(u ), t T where C(u ) does not depend on T. Since is bounded, it follows that for any q 2r + 2. sup u q C(u ) t T We will prove the theorem with an iteration argument, similar to that found in [1]

9 Proof. For p > 1, multiply equation (2.1) by u u p 1 and integrate over, to obtain 9 (2.1) u u p 1 u t dx = a (x, u) u (u u p 1 (x))dx J(x y)u(x) (u u p 1 (x))dydx + J(x y)u(y) (u u p 1 (x))dydx. Since (2.11) and a(x, u) u (u u p 1 )dx = p a (x, u) u p 1 u 2 dx (2.12) u p = with condition (A 2 ), we have (p + 1)2 u p 1 u 2, 4 (2.13) a (x, u) u (u u p 1 )dx 4pc 1 (p + 1) 2 + 4pc 2 (p + 2r + 1) 2 u p dx u p+2r dx. This yields (2.14) 1 d p + 1 dt u p+1 dx + 4pc 1 (p + 1) 2 u p dx J(x y)u(x) (u u p 1 (x))dydx + J(x y)u(y) (u u p 1 (x))dydx. From Cauchy-Schwartz and Young s inequalities we have (2.15) J(x y)u(x) (u u p 1 (x))dydx u p dx + M 2 p u p+1 dx, c 1p (p + 1) 2

10 1 PETER W. BATES for some positive constant M 2 which does not depend on p, and M 1 = sup J(x y)dy. Also we have J(x y) u(y) (u u p 1 (x))dydx (2.16) = p J(x y) u(y) u(x) p 1 u(x) dxdy c 1p (p + 1) 2 u p dx + M 3 p u p+1 dx for some constant M 3 which does not depend on p. Inequalities (2.14)-(2.16) imply (2.17) d dt u p+1 dx + 2pc 1 u p dx C (p + 1) 2 u p+1 dx. (p + 1) Now we need the following Nirenberg-Gagliado inequality, (2.18) where (2.19) D j v L s C 1 D m v a L r v 1 a L q + C 2 v L q, j m a 1, 1 s = j n + a(1 r m n ) + (1 a)1 q. In (2.18), set s = 2, j =, r = 2, m = 1, to get (2.2) v 2 2 C 1 Dv 2a 2 v 2(1 a) q + C 2 v 2 q. Let v = u µ k +1 2, µ k = 2 k, q = 2(µ k 1+1) µ k +1, and (2.21) a = n(2 q) n(2 q) + 2q = n n k. Using Young s inequality this yields u µ k+1 dx ɛ u µ k (2.22) dx + cɛ a 1 a ( u µ k 1+1 dx) If we set p = µ k in (2.17) and plug (2.22) into (2.17), we obtain (2.23) d u µ k+1 dx + 2c 1µ k dt µ k + 1 C(µ k + 1) 2 (ɛ u µ k dx u µ k dx + cɛ a 1 a ( 1 Choosing ɛ = C(µ k c1µ k +1) 2 µ k +1, we have u µ k 1+1 dx) µ k +1 µ k µ k +1 µ k 1 +1 ).

11 11 (2.24) d dt u µ k+1 dx + C 1 (k) u µ k dx C 2 (k)( u µ k 1+1 dx) where C 1 (k) = c 1µ k µ k +1, C 2(k) = C 1 1 a c ( c 1 µ a k µ k +1 ) 1 a (µk + 1) 2 1 a. Choosing ɛ = 1 in (2.22), this and (2.24) also imply d dt u µ k+1 dx + C 1 (k) u µ k+1 dx C 4 (k)( u µ k 1+1 dx) where C 4 (k) = C 2 (k) + c. By Gronwall s inequality, we have u µ k+1 dx u µ k+1 dx + C 4(k) C 1 (k) (sup u µ k 1+1 dx) t (2.25) δ(k)max{m µ k+1, (sup u µ k 1+1 dx) t where δ(k) = c(1 + µ k ) α, α = 2 1 a, and M = sup u. This implies x (2.26) u µ k+1 dx δ(k)max{m µ k+1, (sup u µ k 1+1 dx) t k µ k +1 µ ( δ(k i)) k i +1 max{m µ k+1, (sup t i= Since µ k+1 µ k i +1 < 2i, we have (2.27) and (2.28) µ k +1 µ k +1 µ δ(k)δ(k 1) k 1 +1 µ δ(k 2) k 2 +1 δ(1) µ k +1 2 µ k +1 c 2k 1 (2 α ) k+2k+1 2 µ k 1 +1 µ k k +1. Estimates (2.26)-(2.28) and Lemma 2.2 imply ( u µ k+1 1 µ dx) k +1 (2.29) C 2 2α max{m, sup( t µ k +1 µ k 1 +1 } µ k +1 µ k 1 +1, µ k +1 µ k 1 +1 µ k +1 µ k 1 +1 µ k +1 µ k 1 +1 }, u 2 dx) µ k +1 2 }. u 2 dx) 1 2 } C(u ) where C(u ) does not depend on k. Since this is true for any k, letting k in (2.29), we have and therefore, u C(u ), (2.3) sup u C(u ). t T

12 12 PETER W. BATES Since u C( Q T ), it follows that max Q T u(x, t) C(u ) Remark 2.3. In (2.3), since C(u ) does not depend on T, we also obtain a global bound for u whenever there is global existence of a classical solution. Since max QT u M, after a slight modification of the proof of Theorem 7.2 in Chapter V in [65], using the equivalent form (2.5) we have Theorem 2.4. For any solution u C 2,1 ( Q T ) of equation (2.1) (2.3) having max QT u C, one has the estimates (2.31) max u K 1, u (1+δ) Q Q T K 2, T where constants K 1, K 2, and δ depend only on C, u C 2 ( ) is a Hölder norm in [65]. and, (1+δ) Q T In (2.5), setting v(x, t) = u(x, t) u (x), we obtain the equivalent form v t = ã(x, v, u ) v + b(x, v, v, u ) in, t >, (2.32) ã(x, v, u ) v n + ψ(x, v, u ) = on, t >, where and v(x, ) =, ã(x, v, u ) = a(x, v + u ), b(x, v, v, u ) = a(x, v + u ) u + b(x, v + u, (v + u )), ψ(x, v, u ) = a(x) n (v(x, t) + u (x)) + ã(x, v, u ) u n J(x y) (v(y, t) + u (y))dy. n Since (2.4) implies ψ(x,, u ) =, the compatibility condition for (2.32) is also satisfied. Denote Lv = v t ã(x, v, u ) v b(x, v, v, u ), and L v = v t c 1 v, where c 1 is the constant in condition (A 2 ).

13 13 Consider the following family of problems: (2.33) λlv + (1 λ)l v = in Q T, λ(ã(x, v, u ) v n + ψ(x, v, u )) + (1 λ)(c 1 ( v n )) = on [, T ], v(x, ) =. Lemma 2.5. If v(x, t, λ) C 2,1 ( Q T ) is a solution of (2.33), then (2.34) where K does not depend on λ. max Q T v(x, t, λ) K, Proof. Since λã(x, v, u ) + (1 λ)c 1 λc 1 + (1 λ)c 1 = c 1 >, the terms in (2.33) also satisfy (A 1 ) (A 2 ) and so (2.34) follows from Theorem 2.1. Consequently one may also conclude from Lemma 2.5 and Theorem 2.4 that: Lemma 2.6. If v(x, t, λ) C 2,1 ( Q T ) is a solution of equation (2.33), then (2.35) max Q T v x (x, t, λ) K 1, v(x, t, λ) (1+δ) Q T K 2, where constants K 1, K 2, and δ do not depend on λ. Define a Banach space 1+β 1+β, X = {v(x, t) C 2 ( QT ) : v(x, ) = } with the usual Hölder norm. For any function w X satisfying conditions max QT w M and max QT w x M 1, we consider the following linear problem (2.36) v t (λã(x, w, u ) + (1 λ)c 1 ) v + λ b(x, w, w, u ) = in Q T, λ(ã(x, w, u ) v n + ψ(x, v w, u )) + (1 λ)c 1 n = on [, T ], v(x, ) =. 2+β 2+β, It is clear that there exists a unique solution v(x, t, λ) C 2 ( Q T ) of (2.36). Define T (w, λ) by v(x, t, λ) = T (w, λ). It is fairly straightforward to show that for w being in a bounded set of X, T (w, λ) is uniformly continuous in λ, and that for any fixed λ, T (x, λ) is continuous in X. 2+β, 2+β 1+β, 1+β 2 ( Q T ) is compact, we have that for any Since C 2 ( Q T ) C fixed λ, T (w, λ) is a compact transformation. The Leray-Schauder Fixed Point Theorem (see, e.g., [65]) gives the existence of a solution v(x, t) of (2.32), and therefore:

14 14 PETER W. BATES Theorem 2.7. For u C 2+β ( ) for some β > satisfying the boundary condition (2.4), there exists a solution u to (2.1) (2.3) with u C To continue with the well-posedness question, we have 2+β, 2+β 2 ( Q T ). Theorem 2.8. (Uniqueness and continuous dependence on initial data) If u 1 (x, t) and u 2 (x, t) are two solutions corresponding initial data u 1 (x) and u 2 (x) of equation (2.1) (2.3), then for some C depending only on T, (2.37) sup t T u 1 u 2 dx C u 1 u 2 dx. Proof. For any θ C 2,1 ( Q T ) with θ n = on, we have (2.38) u i (x, τ)θ(x, τ)dx = τ u i (x, )θ(x, )dx + τ where B(x, u) = a(x)u + f(u). Hence, + θ J u i dxdt + τ (u i θ t + B(x, u i ) θ)dxdt θ J n u idxdt, (2.39) (u 1 u 2 )θ(x, τ)dx = (u 1 u 2 )θ(x, )dx τ τ + (u 1 u 2 )(θ t + H θ)dxdt + θ J (u 1 u 2 )dxdt τ + θ J n (u 1 u 2 )dxdt, where H(x, t) = B(x,u 1 ) B(x,u 2 ) u 1 u 2 for u 1 u 2 B(x,u 1 ) u for u 1 = u 2. Let θ be the solution to the final value problem (2.4) θ t = H(x, t) θ + βθ in, t τ, θ n = on, θ(x, τ) = h(x), where h(x) C (), h 1 and β > is a constant. By the comparison theorem, we have θ e β(t τ).

15 15 Therefore, from (2.39) we have (u 1 u 2 )hdx τ (2.41) = (u 1 u 2 )θ(x, )dx + τ + θ J (u 1 u 2 )dxdt + Hence, (2.42) (u 1 u 2 )βθdxdt τ θ J n (u 1 u 2 )dxdt. (u 1 u 2 )hdx τ u 1 u 2 e βτ dx + u 1 u 2 βe β(t τ) dxdt τ τ + C 1 u 1 u 2 e β(t τ) dxdt + C 2 u 1 u 2 e β(t τ) dxdt. Letting β and h sign(u 1 u 2 ) + in (2.42), we have τ (2.43) (u 1 u 2 ) + dx u 1 u 2 dx + C 3 u 1 u 2 dxdt. Interchanging u 1 and u 2 gives (2.44) u 1 u 2 dx By Gronwall s inequality, (2.44) yields τ u 1 u 2 dx + C 3 u 1 u 2 dxdt. (2.45) u 1 u 2 dx C(T ) u 1 u 2 dx. Remark 2.9. If u (x) L (), we can consider weak solutions as follows: Define X = {f(x) C () g(x) = J(x y)f(y)dy, g(x) = } B = Closure of X in the L 2 norm Definition 2.1. A weak solution of (2.1) (2.3) is a function u C([, T ], L 2 ()) L (Q T ) L 2 ([, T ], H 1 ()), u t L 2 ([, T ], H 1 ()), h(x, u) L 2 ((, T ), L 2 ()) such that

16 16 PETER W. BATES (2.46) < u t (x, t), ψ(x) > + h(x, u) ψ(x)dx ( J u(, s)) ψ(x)dx = for all ψ H 1 () and a.e. time t T, where h(x, u) = a(x)u + f(u), a(x) = J(x y)dy, and u(x, (2.47) ) = u(x). Theorem If (A 1 ), (A 2 ), and (A 3 ) are satisfied and u L () B, then there exists a unique weak solution u of (2.1) (2.3) The proof is as follows: Since u L () B, there exists a sequence u (k) X such that (2.48) u (k) u L 2, u (k) < C, where C does not depend on k. Consider equation (2.1) (2.3) with initial data u (k). There exists a unique classical solution u(k). By the energy estimate and other a priori bounds, one can find a subsequence and a weak limit u such that (2.49) u (k) u in h(x, u (k) ) h(x, u) in and u satisfies equation (2.46). L 2 ((, T ), H 1 ()), L 2 ((, T ), H 1 ()), u (k) t u t in L 2 ((, T ), H 1 ()), Now we turn to discussing the long-term behavior of solutions in the L p norm We establish a nonlinear version of the P oincaré inequality. Proposition Let R n be smooth and bounded. For p 1, there is a constant C() such that for all u W 1,2p () with u = (2.5) u 2p dx C() u p 2 dx. Proof. If (2.5) is not true, there exists a sequence {u k } W 1,2p () such that (2.51) u k =, u k 2p dx > k u k p 2 dx.

17 u k If w k = u k 2p, then it follows that (2.52) w k =, w k 2p dx = 1, w k p 2 dx < 1 k. Therefore, there exists a subsequence (still denoted by { w k p }) and w H 1 () such that (2.53) w k p w in H 1 and w k p w in L 2. Since w k p 2 dx 1 k, for any ϕ C (2.54) for i=1,,n. Therefore, (2.55) w k p x i ϕdx w x i ϕdx = (), we have for i=1,,n and ϕ C (). So w = a.e in, and w is constant in. By taking a subsequence, (2.52) and (2.53) yield 17 (2.56) So, we have w = ( 1 ) 1 2, and wk p ( 1 ) 1 2 a.e in. (2.57) w k ( 1 ) 1 2p a.e in. Since w k =, there exists a unique solution ϕ k to ϕ = w k in, ϕ (2.58) n = on, ϕdx =. From (2.58), we obtain (2.59) ϕ k 2 = w k ϕ k w k L 2 ϕ k L 2. Since ϕ kdx =, by P oincaré s inequality, ϕ k L 2 c ϕ k L 2, therefore (2.58) and (2.59) imply (2.6) and (2.61) ϕ k L 2 c w k L 2 ( w k p 1 w k ) ϕ k dx = w k p+1 dx.

18 18 PETER W. BATES Since (2.62) we have ( w k p 1 w k ) = p w k p 1 w k, (2.63) ( w k p 1 w k ) = p w k p 1 w k = w k p. Hence, from (2.61), we have w k p+1 dx = (2.64) as k, by (2.52) and (2.6). Hence, along a subsequence, i.e, (2.65) This contradicts (2.57). ( w k p 1 w k ) ϕ k dx w k p ϕ k dx w k p L 2 ϕ k L 2 w k p+1 a.e in, w k a.e in. Remark After this was complete, we became aware of a similar result by Alikakos and Rostamian in [4] but we include our result for completeness. The next step is to establish the existence of an absorbing set in L q+1 for all q > 1. This is done by first writing the equation in terms of v = u u, multiplying that equation by v v q 1, and integrating. Then one uses Proposition 2.12 and a uniform Gronwall inequality found in [75] to obtain the following: Proposition Let α < ( c 1 2r, where c 1 and c 2 are the constants in assumption (A 2 ), and let ū = 1 u dx. If u(x) is a solution of (2.1)- (2.3), and ū α, then for any q > 1, we have (2.66) u ū q+1 dx < C 1 + ( C 2rt q+1 ) 2r q + 1 c 2 ) 1 where C 1 depends on α and q, and C 2 depends on q. Consequently, one has for any solution of (2.1)-(2.3) with 1 u dx = ū α, there exists a time t (α, q) such that (2.67) u q+1 < µ, for all t > t (α, q),

19 19 where C 3 (q) µ > ( C 4 (q, α ) ) 1 q+2r+1 + α 1 q+1. We have shown that in L p there exists a local absorbing set in the sense that if u is not too large, the solution enters a fixed bounded set in the affine space ū + L p in finite time (note that ū = 1 u is conserved by the evolution). Now we consider the long term behavior of the solution in the H 1 norm. In this case, we do not need any restriction on u. Note that (A 2 ) implies f(u)u c 5 u 2r+2 c 6 for some constants c 5 and c 6. We make additional assumptions on the nonlinearity, (A 4 ) f(u) c 7 u 2r+1 + c 8, (A 5 ) F (u) = u f(s)ds c 9 u 2r+2 + c 1, and c 5 > c 9. Remark (A 2 ), (A 4 ), and (A 5 ) hold for f(u) = c u 2r u+ lower terms. Denote ψ = 1 ψdx, write ϕ = ψ ψ. For ϕ L 2 (), satisfying ϕ =, we consider the following equation: (2.68) θ = ϕ θ n = θ = The equation (2.68) has a unique solution θ := ( ) 1 (ϕ). Denote ϕ 1 = ( ( ) 1 (ϕ)ϕdx) 1 2. This is a continuous norm on L 2 (). Since ū = ū is constant, we may write the equation as (2.69) (u ū) t = K(u), where K(u) = J(x y)dyu(x) J(x y)u(y)dy + f(u). Applying the operator ( ) 1 to both sides of equation (2.69), we obtain (2.7) d( ) 1 (u ū) dt + K(u) =. Taking the scalar product with u ū in L 2 (), we have (2.71) 1 d 2 dt u ū (K(u), u ū) =. From condition (A 2 ) (A 5 ), we have

20 2 PETER W. BATES (2.72) (K(u), u ū) 1 J(x y)(u(x) u(y)) 2 dxdy + c 5 2 ɛ u 2r+2 dx c(ū, ɛ) u 2r+2 dx for any ɛ >. Choosing ɛ = c 5 c 9, we have (2.73) (K(u), u ū) 1 J(x y)(u(x) u(y)) 2 dxdy + c 9 2 E(u) c(ū) = E(u) c(ū ). Also from (2.73), we have (2.74) (K(u), u ū) c u 2r+2 dx c(ū ) for some positive constants c and c(ū ). Since. 1 is a continuous norm on L 2 (), we have u 2r+2 dx c(ū) (2.75) Therefore, u ū 1 C u ū 2. (2.76) u ū 1 C u ū 2 C u 2r+2 + C(ū ) for some positive constants C and C(ū ). From (2.71), (2.74), and (2.76), it follows that d (2.77) dt u ū C u ū 2r+2 1 C(ū ). By the uniform Gronwall inequality mentioned above, we obtain u ū 2 1 ( C(ū ) C ) 1 1 (2.78) r+1 + (C(r)t) Thus, we have proved: Theorem There exists M(ū ) such that for any ρ > M(ū ) 1 2r+2, there exists a time t such that (2.79) u ū 1 ρ, t t. r.

21 21 From (2.71) and (2.72), we also obtain 1 d (2.8) 2 dt u ū E(u) c(ū ). Integrating from t to t + 1, then (2.79) implies (2.81) t+1 t E(u(s))ds c (ū ) c(ū ) + ρ2 2 for t t. Since E(u(t)) is decreasing, (2.81) implies (2.82) for t t + 1. Since, from (2.6), (2.83) E(u(t)) 1 4 c 3 E(u(t)) c (ū ) J(x y)(u(x) u(y)) 2 dxdy + u 2r+2 c 4, F (u)dx inequalities (2.82) and (2.83) yield (2.84) for t t. u 2r+2 c (ū ) Corollary There exists c (ū ) > M(ū ) 1 2r+2 such that for any ρ > c (ū ), there exists a time t such that (2.85) u r+1 c (ū ) for t t. Next we estimate u 2. Denote h(x, u) = a(x)u + f(u), multiplying (2.1) by h(x, u) and integrating over, we have (2.86) h(x, u)u t + h(x, u) 2 = J u h(x, u). Since (2.87) and (2.88) h(x, u)u t = (a(x)u + f(u))u t = t [1 2 a(x)u2 + F (u)], J u h(x, u) c u h(x, u) 2 2, equation (2.86) yields (2.89) d dt [ 1 2 a(x)u2 + F (u)] h(x, u) 2 c u 2 2.

22 22 PETER W. BATES Integrate (2.89) from t to t + 1, and use assumption (A 2 ) and Corollary 2.17, to obtain t+1 (2.9) h(x, u) 2 c t for some constant c and all t t. Multiply (2.1) by h(x, u) t and integrate on, to obtain (2.91) Since h(x, u) t u t + h(x, u) h(x, u) t = J u h(x, u) t. (2.92) h(x, u) t u t = a(x)u 2 t + f (u)u 2 t c 1 u 2 t, h(x, u) h(x, u) t = 1 d h(x, u) 2, 2 dt and J u h(x, u) t = d J u h(x, u) dt J u t h(x, u), we have (2.93) c 1 u t d h(x, u) 2 2 dt d J u h(x, u) dt J u t h(x, u). Estimate (2.93) with the Cauchy-Schwartz, and Young s inequalities imply d h(x, u) 2 d (2.94) 2 J u h(x, u) + γ h(x, u) 2 dt dt for some constant γ >. For t < s < t + 1, multiplying (2.94) by e γ(t s), we have d ds [eγ(t s) h(x, u) 2 ] e γ(t s) d (2.95) 2 J u h(x, u). ds Integrating (2.95) between s and t + 1, we obtain e γ h(x, u(x, t + 1)) 2 e γ(t s) h(x, u(x, s)) 2 (2.96) t+1 e γ(t µ) d 2 J u(, µ) h(x, u(x, µ))dxdµ. dµ s

23 Since (2.97) t+1 s = e γ(t µ) e γ(t µ) d 2 J u(, µ) h(x, u(x, µ))dxdµ dµ t+1 s = I 1 + I 2. 2 J u(, µ) h(x, u(x, µ))dx t+1 s ( γ)e γ(t µ) These may be individually estimated yielding 2 J u(, µ) h(x, u(x, µ))dxdµ (2.98) e γ h(x, u(x, t + 1)) 2 e γ(t s) h(x, u(x, s)) 2 e γ h(x, u(x, t + 1)) 2 + C u(x, t + 1) t+1 h(x, u(x, s)) 2 + C u(x, s) 2 + C [ h(x, u(x, µ)) 2 + u(x, µ) 2 ]dµ. s Therefore, (2.99) e γ h(x, u(x, t + 1)) 2 2 e γ(t s) h(x, u(x, s)) 2 + C + C u(x, s) 2 + C t+1 s u(x, t + 1) 2 + [ h(x, u(x, µ)) 2 + h(x, u(x, s)) 2 u(x, µ) 2 ]dµ. Integrating (2.99) from t to t + 1 with respect to s, we have (2.1) e γ 2 + t+1 t t+1 + C h(x, u(x, t + 1)) 2 dx t t+1 t h(x, u(x, s)) 2 dxds + C h(x, u(x, s)) 2 dxds + C (µ t)[ t+1 h(x, u(x, µ)) 2 dx + By (2.85) and (2.9), estimate (2.1) yields (2.11) h(x, u(x, t + 1)) 2 dx C(ū ) for t t (ū ) and some C(ū ) >. t u(x, t + 1) 2 dx u(x, s) 2 dxds u(x, µ) 2 dx]dµ. 23

24 24 PETER W. BATES Since (2.12) h(x, u(x, t + 1)) = (a(x) + f (u(t + 1))) u(x, t + 1) u(x, t + 1) a(x), we have (2.13) h(x, u(x, t + 1)) 2 1 (a(x) + f (u(t + 1))) 2 u(x, t + 1) 2 2 u(x, t + 1) a(x) c2 1 u(x, t + 1) 2 D(ū ) for t t (ū ) and some constant D(ū ). Estimates (2.11) and (2.13) imply (2.14) u(x, t + 1) 2 G(ū ), for t t (ū ) and G(ū ) >. Thus, we have Theorem There exists a time t (ū ) such that (2.15) u H 1 c(ū ) for t t (ū ). Remark [75] gives a similar result for the Cahn-Hilliard equation. This boundedness gives weak convergence of subsequences as t n but more is true, as can be demonstrated by calculations similar to the foregoing: Theorem 2.2. If u is a solution of (2.1)-(2.3), and Q(u) = ( J(x y)dy)u(x) J u(x) + f(u(x)), then there exist a sequence {t k } and u such that u(t k ) u weakly in H 1 (2.16) Q(u(t k )) Q(u ) weakly in H 1 and Q(u ) is a constant, i.e. u is a steady state solution of (2.1)-(2.3). We may also use the above techniques for the following integrodifferential equation that may be derived from interacting particle systems with Kawasaki dynamics (2.17) u t = (u tanhβj u) in (u tanhβj u) n = on u(x, ) = u (x) where β is a constant, J is a smooth function.

25 Wellposedness and regularity of solutions is established along the lines used for (2.1)-(2.3) with the usual smoothness assumptions on J, f,, and the initial data. Note that the average of u is constant in time and one can show that there is an absorbing set in every constant mass affine subspace of H 1. Returning to the nonlocal Cahn-Hilliard equation (2.1), we may also append the homogeneous Dirichlet boundary condition (2.18) u(x) = for x. While we no longer have conservation of the integral, this boundary condition is strongly dissipative and so we expect results similar to those above. In particular with condition (A 2 ) D There exists c 1 > such that a(x, u) a(x) + f (u) c 1, one can prove Proposition Assume (A 1 ), (A 2 ) D, and (A 3 ). If u(x, t) C( Q T ) C 2,1 (Q T ) is a solution of (2.1),(2.18), with initial data u then 25 (2.19) max Q T u C(, T, u ) for some positive constant C(, T, u ). This is proved by letting u(x, t) = v(x, t)e σt for appropriate choice of σ, multiplying the v-equation by v to obtain an equation for v 2, and applying a maximum principle. A result similar to Theorem 2.4 gives gradient and Hölder-(1+α) bounds on the solution for small α >. Then it is straightforward to apply the Schauder Fixed Point Theorem to establish the existence of a classical solution. Under the assumption (A 2 ) D, equation (2.1) is a nondegenerate parabolic equation. We may also consider the degenerate case. Consider the following equation with u L () (2.11) where u t = (h(x, u)) u(y) J(x y)dy in Q T u = on S T u(x, ) = u (x), h(x, u) = a(x)u(x) + f(u). Instead of nondegeneracy condition (A 2 ) D, we assume: (B 1 ) For every fixed x, h(x, ) =, and h(x,u) u d 1 u r 1 for some positive constants r 1 and d 1.

26 26 PETER W. BATES Definition A generalized solution of (2.11) is a function u C([, T ] : L 1 ()) L (Q T ) such that (2.111) u(x, t)ψ(x, t)dx u(x, t)ψ s (x, s)dxds = h(x, u) ψ(x, s)dxds Q t Q t ( J u(, s))ψ(x, s)dxds + u(x, )ψ(x, )dx Q t for all ψ C 2,1 ( Q T ) such that ψ(x, t) = for x and t T, and (2.112) u(x, ) = u (x). We first prove the uniqueness. Proposition Let u 1, u 2 be two solutions of equation (2.11) with initial data u 1, u 2 L (), then u 1 (τ) u 2 (τ) L 1 () C(T ) u 1 u 2 L 1 () for each τ (, T ), and some constant C(T ). Proof. For any τ (, T ), and ψ C 2,1 ( Q τ ) with ψ = for < t < τ, after multiplying (2.11) by ψ and integrating over (, τ), we have (2.113) u i (x, τ)ψ(x, τ)dx = τ u i (x, )ψ(x, )dx + τ + ( J u i )ψdxdt. (u i ψ t + h(x, u i ) ψ)dxdt Setting z = u 1 u 2 and z = u 1 u 2, equation (2.113) gives (2.114) z(x, τ)ψ(x, τ)dx = where + τ b(x, t) = z (x)ψ(x, )dx z(ψ t + b(x, t) ψ)dxdt + τ { h(x,u1 ) h(x,u 2 ) u 1 u 2 for u 1 u 2, h u (x, u 1 ) for u 1 = u 2. Follow the idea in [6], we consider problem: ( J z)ψdxdt,

27 27 (2.115) ψ t = b ψ + νψ in, < t < τ, ψ = on, < t < τ, ψ(x, τ) = g(x), where g(x) C (), g 1, and ν > is constant. Since b just belongs to L (Q T ) and may be equal to zero, we perturb to get a nondegenerate equation, by setting b n = ρ n b + 1 n, where ρ n is a mollifier in R n, and τ (ρ n b b) 2 dxdt 1. Consider n 2 (2.116) ψ t = b n ψ + νψ in, < t < τ, ψ = on, < t < τ, ψ(x, τ) = g(x). Since b n 1 n, the equation is a nondegenerate parabolic equation, and so there exists a solution ψ n C 2,1 ( Q τ ). The following, whose proof we omit, is easily established. Lemma The solution of (2.116) has the following properties (i) ψ n e ν(t τ), τ (ii) b n (ψ n ) 2 dxdt C, (iii) ψ n 2 dx C, sup t τ where the constant C depends only on g. Replacing ψ by ψ n in (2.114), and using (2.116) we obtain (2.117) z(x, τ)g(x)dx = Since τ C( τ τ z(b b n ) ψ n dxdt z(x, )ψ n ()dx + ( J z + νz)ψ n dxdt. Q τ z(b b n ) ψ n dxdt (b b n ) 2 dxdt) 1 2 ( b n C n, τ b n ψ n 2 dxdt) 1 2

28 28 PETER W. BATES equation (2.117) implies z(x, τ)g(x)dx (2.118) z(x, ) e ν(t τ) dx + J z + νz e ν(t τ) dxdt. Q τ Letting ν and g(x) signz + (x, τ) in (2.118), we have (2.119) (u 1 u 2 ) + dx Interchanging u 1 and u 2 yields (2.12) u 2 u 1 dx u 1 u 2 dx + J z dxdt. Q τ u 2 u 1 dx + C u 2 u 1 dxdt. Q τ (2.12) and Gronwall s inequality imply the conclusion. Remark Since every classical solution is also a weak solution, this also proves the uniqueness and continuous dependence on initial values for classical solutions. To prove the existence of a solution to (2.111), we consider the regularized problem and take u C 2+α ( ) for some α >, with u = : (2.121) where u t = (hɛ (x, u)) J(x y)u(y)dy in Q T, u = on S T, u(x, ) = u (x), h ɛ (x, u) = a(x)u(x) + f(u) + ɛu. 2+α 2+α, We have shown that there exists a classical solution u ɛ (x, t) C 2 ( Q T ). It is easy to show that these solutions are uniformly bounded on Q T. Using the growth conditions and Arzela-Ascoli s lemma, one can then prove Theorem For any T > and u L (), if conditions (A 1 ), (B 1 ), and (A 3 ) are satisfied, then there exists a unique function u C([, T ], L 1 ()) L (Q T ) which satisfies equation (2.111). Results concerning the long-term behavior of solutions to the Dirichlet problem follow from similar ideas introduced for the case of no-flus boundary conditions but this time a nonlinear version of the P oincaré inequality is not used. In order to prove the existence of an absorbing set, instead of (A 2 ) D, we assume the original (A 2 ) and

29 (A 4 ) There exist positive constants c 3 and c 4 such that a(x, u) c 3 u r + c 4. First one establishes L p bounds for solutions by using a Gronwall inequality after multiplying the equation by a power of u and integrating, performing some tedious manipulations. Then one proves Proposition If u L (), then (2.122) sup u C(u ). t Also, gradient bounds may be obtained using the above and some messy calculations: Theorem Assume that u is a solution of (2.1), (2.18) and conditions (A 1 ) (A 4 ) are satisfied. There exists t > such that if t t then (2.123) sup t t u 2 < C, where constant C does not depend on initial data. If we restrict our attention to one space dimension where better embedding theorems are in force, one can then prove: Theorem For n = 1, if conditions (A 1 ) (A 4 ) are satisfied, then the semigroup associated with (2.1) with Dirichlet boundary conditions possesses an attractor A H 1 () X which is maximal and compact. 3. Nonlocal Phase-Field System We now turn to the system where the temperature evolves and the order parameter represents local solidification, partially driven by temperature and phase change in turn producing or absorbing heat energy, thus driving temperature. The following presents some results reported in [19]. As outlined above, this system has the form: (3.1) u t = J u u J(x y)dy f(u) + lθ, (3.2) (θ + lu) t = θ, which is complemented by the initial and boundary conditions (3.3) u(, x) = u (x), θ(, x) = θ (x), θ (3.4) n =, where T > and R n is a bounded domain. We are interested in the well posedness of this initial and boundary value problem. In order to prove the existence, we make the following assumptions (P 1 ) M sup J(x y) dy < and f C(R). 29

30 3 PETER W. BATES (P 2 ) There exist c 1 >, c 2 >, c 3 >, c 4 > and r > 2 such that f(u)u c 1 u r c 2 u, and f(u) c 3 u r 1 + c 4. Note that (P 2 ) implies (3.5) F (u) = u f(s)ds c 5 u r c 6 u for some positive constants c 5 and c 6. We prove the existence of a solution to (3.1)-(3.4) by the method of successive approximation. Define θ () (t, x) := θ (x) and for k 1 (u (k), θ (k) ) iteratively to be solutions to the system (3.6) u (k) t = (3.7) θ (k) t J(x y)u (k) (y)dy J(x y)dyu (k) (x) f(u (k) ) + lθ (k 1), θ (k) + θ (k) = lu (k) t + θ (k 1) in (, T ), with initial and boundary conditions (3.8) (3.9) u (k) (, x) = u (x), θ (k) (, x) = θ (x), θ (k) n =. Lemma 3.1. With k = 1, for any T >, if u L (), and θ H 1 L (), then there exists a unique solution (u, θ) to system (3.6) -(3.9). Furthermore, u (1), u (1) t L ((, T ), L ()) and θ (1) L ((, T ), L ()) L 2 ((, T ), H 2 ()). Proof. Since the right hand side of equation (3.1) is locally Lipschitz continuous in L ((, T ), L ()), local existence follows from standard ODE theory. In order to prove the global existence, we prove global boundedness of the solutions. For any p > 1, multiplying equation (3.1) by u (1) p 1 u and integrating over, we obtain 1 d u (1) p+1 dx + f(u (1) ) u (1) p 1 udx p + 1 dt (3.1) = J(x y)u (1) (y) u (1) p 1 u (1) dxdy J(x y)u (1) (x) u (1) p 1 u (1) dxdy + l θ () u (1) p 1 udx. Using Holder s and Young s inequalities and conditions (P 1 ) and (P 2 ), we have 1 d u (1) p+1 dx + C u (1) p+r 1 udx (3.11) p + 1 dt C(p)C p+1 1,

31 where C 1 is a constant independent of p and lim p C(p) 1 p+1 C 2 with C 2 independent of p. Using the uniform Gronwall inequality and (3.11), we have (3.12) Therefore, (3.13) Letting p, we have (3.14) u (1) p+1 p+1 (C(p)Cp+1 1 ) p+1 p+r 1 + (C(r 2)t) (p+1) r 2. u (1) p+1 C(p) 1 p+1 (C 1 ) p+1 p+r 1 + (C(r 2)t) 1 r 2. u (1) C. for some constant C. Also from condition (P 2 ) and equation (3.6), we have (3.15) u (1) t C. Since equation (3.7) is a linear parabolic equation, by inequality (3.15) and standard parabolic theory, we have θ (1) L ((, T ), L ()) L 2 ((, T ), H 2 ()). By induction, there exist unique solution (u (k), θ (k) ) of system (3.6)-(3.8). Furthermore, u (k), u (k) t L ((, T ), L ()) and θ (k) L ((, T ), L ()) L 2 ((, T ), H 2 ()) for every k. Now we prove that there exists a uniform bound for u (k), u (k) t and θ (k). Multiplying equation (3.7) by θ (k) p 1 θ (k) (x) for p > n 2, and integrating over, we have (3.16) θ (k) p 1 θ (k) θ (k) t dx + ( θ (k) p 1 θ (k) ) θ (k) dx + θ (k) p+1 dx = l J(x y)u (k) (y) θ (k) p 1 θ (k) dydx + l f(u (k) ) θ (k) p 1 θ (k) dx + l J(x y)u (k) (x) θ (k) p 1 θ (k) dydx + (1 l 2 ) θ (k) p 1 θ (k) θ (k 1) dx. Since (3.17) θ p = (p + 1)2 θ p 1 θ 2 = 4 using Holder s and Young s inequalities, we obtain (3.18) 1 d θ (k) p+1 dx + 4p p + 1 dt (p + 1) 2 c 1 (l, p) u (k) p+1 dx + c 2 (l, p) (p + 1)2 ( θ p 1 θ) θ, 4p θ (k) p dx + 1 θ (k) p+1 dx 2 θ (k 1) p+1 dx + f(u (k) ) p+1 dx for some positive constants c 1 (l, p) and c 2 (l, p) which depend only on p and l. 31

32 32 PETER W. BATES Multiplying equation (3.6) by u (k) (r 1)p 1 u (k), and integrating over, we obtain (3.19) 1 d u (k) (r 1)p+1 dx + f(u (k) ) u (k) (r 1)p 1 u (k) dx (r 1)p + 1 dt = J(x y)u (k) (y) u (k) (r 1)p 1 u (k) dxdy + l θ (k 1) u (k) (r 1)p 1 u (k) dx J(x y)u (k) (x) u (k) (r 1)p 1 u (k) dxdy. Condition (A 2 ) implies (3.2) and (3.21) f(u) u (r 1)p 1 u c 1 u (r 1)(p+1) c 2 u (r 1)p f(u) p+1 c 7 u (r 1)(p+1) + c 8 for some positive constants c 7 and c 8. From equation (3.19), inequality (3.2), Holder s and Young s inequalities, we have (3.22) 1 d (r 1)p + 1 dt u (k) (r 1)p+1 dx + c 1 u (k) (r 1)(p+1) dx 2 c(r, p) + c 1 (r, p, l) θ (k 1) p+1 dx for some positive constants c(r, p) and c 1 (r, p, l). Integrating (3.22) from to t, we obtain (3.23) 1 (r 1)p + 1 u (k) (r 1)p+1 dx + c 1 2 c(r, p)t + c 1 (r, p, l) t t c(u, T, r, p) + c 1 (r, p, l) t u (k) (r 1)(p+1) dx θ (k 1) p+1 dx + θ (k 1) p+1 dx u (r 1)p+1 for some positive constants c(u, T, r, p) and c 1 (r, p, l). Integrating inequality (3.18) from to t, using (3.21) and (3.23), we have t (3.24) θ (k) p+1 dx c(u, θ, p, r, l, T )(1 + θ (k 1) p+1 dxds) for some positive constant c(u, θ, p, r, l, T ) which does not depend on k. By induction, we have (3.25) θ (k) p+1 dx ce ct for some positive constant c which does not depend on k.

33 Similarly from inequalities (3.22) and (3.25), we also have (3.26) u (k) p+1 dx C, and (3.27) f(u (k) ) p+1 dx C for some positive constant C which does not depend on k. Equation (3.6), inequalities (3.25)-(3.27), and Young s inequality imply (3.28) u (k) t p+1 dx C for some positive constant C which does not depend on k. This implies lu (k) t + θ (k 1) L p+1 ((, T ), L p+1 ()) and (3.29) lu (k) t + θ (k 1) p+1 C for some positive constant C which does not depend on k. Applying standard parabolic estimates to equation (3.7), and using inequality (3.28), we have 33 (3.3) θ (k) C. Multiplying equation (3.7) by θt k, and integrating equation (3.7) over, using Holder and Young s inequalities and (3.3), we have T (3.31) θ (k) t 2 dxdt C for some constant C which does not depend on k. Equation (3.7), inequalities (3.28), (3.3), and (3.31) yield T (3.32) θ (k) 2 dxdt C for some constant C which does not depend on k. Since θ (k) C, using a similar argument to that in the proof of Lemma 3.1, we have (3.33) u (k) C, and (3.34) u (k) t C for some constant C which does not depend on k.

34 34 PETER W. BATES Next we prove the convergence of {θ (k) } in C([, T ], L 2 ()). From equation (3.7), we have (3.35) (θ (k+1) θ (k) ) t (θ (k+1) θ (k) ) + (θ (k+1) θ (k) ) = l(u (k+1) u (k) ) t + (θ (k) θ (k 1) ). Multiplying equation (3.35) by (θ (k+1) θ (k) ), and integrating over, using Holder s and Young s inequalities, we have (3.36) 1 d 2 dt θ (k+1) θ (k) 2 dx + (θ (k+1) θ (k) ) 2 dx + 1 (θ (k+1) θ (k) ) 2 2 l 2 u (k+1) t u (k) t 2 dx + θ (k) θ (k 1) 2 dx. Since u (k) C, from equation (3.6), and condition (P 2 ), we have (3.37) u (k+1) u (k) 2 dx C(T ) θ (k) θ (k 1) 2 dx, and (3.38) f(u (k+1) ) f(u (k) ) 2 dx = f (λu (k+1) + (1 λ)u (k) )(u (k+1) u (k) ) 2 dx C(T ) u (k+1) u (k) 2 dx. Therefore, equation (3.6), and inequalities (3.37)-(3.38) imply (3.39) u (k+1) t u (k) t 2 dx 4 J(x y)(u (k+1) u (k) )dy 2 dx + 4 ( J(x y)dy) 2 (u (k+1) u (k) ) 2 dx + 4 (f(u (k+1) ) f(u (k) ) 2 dx + 4 (θ (k) θ (k 1) ) 2 dx C 1 (T ) θ (k) θ (k 1) 2 dx for some positive constant C 1 (T ) which does not depend on k. Inequalities (3.36)-(3.39) yield d (3.4) θ (k+1) θ (k) 2 dx C(T ) θ (k) θ (k 1) 2 dx dt for some positive constant C(T ) which does not depend on k. By induction, this implies t (3.41) θ (k+1) θ (k) 2 dx (ct)(k 1) θ 1 θ dxds. (k 1)!

35 So θ (k) is a Cauchy sequence in C([, T ], L 2 ()). Therefore, there exists θ C([, T ], L 2 ()) such that θ (k) θ in C([, T ], L 2 ()). From (3.3)- (3.32), we have (3.42) (3.43) (3.44) T T Also from (3.33), (3.37)-(3.39), we have (3.45) θ C, θ 2 dxdt C, θ t 2 dxdt C. u (k) u in C([, T ], L 2 ()), 35 (3.46) u (k) t u t in C([, T ], L 2 ()), (3.47) f(u (k) ) f(u) in C([, T ], L 2 ()). Therefore, letting k in equation (3.6), we have (3.48) u t = J(x y)u(y)dy J(x y)dyu(x) f(u) + lθ for t > and a.e. x. Since u (k) t u t, θ (k) t θ t, θ (k) θ in L 2 ((, T ), L 2 ()), letting k in the weak form of equation (3.7), we have T T (3.49) (lu t + θ t )ξ(t, x)dxdt = θξ(t, x)dxdt for ξ(t, x) L 2 ((, T ), L 2 ()). Since it is true of θ (k), we also have T (3.5) η(t)( θϕ + θ ϕ)dxdt = for any ϕ W 1,2 () and η L 2 (, T ). This implies θ n = a.e on (, T ). Also we have (3.51) θ(, x) θ 2 dx 3( θ(, x) θ(t, x) 2 dx + θ(t, x) θ (k) (t, x) 2 dx + θ (k) (t, x) θ 2 dx). Since θ (k) (t, x) θ in C([, T ], L 2 ()), and since θ (k) (t, x) and θ(t, x) are continuous with respect to t in L 2 (), by taking k arbitrarily large we can see that θ(, x) = θ a.e. in. Similarly, u(, x) = u a.e. in. Equations (3.48)-(3.51) imply that u and θ are solutions of system (3.1)- (3.4) in a weak sense.

36 36 PETER W. BATES To prove uniqueness and continuous dependence on initial data, let θ i L () W 1,2 (), u i L (), and for R >, θ i L R, u i L R, where i = 1, 2. Let u i and θ i be solutions corresponding to initial data u i and θ i, then we have θ i L C(T, R), and u i L C(T, R). Denote v = u 1 u 2, w = θ 1 θ 2. We have (3.52) v t = (3.53) (w + lv) t = w J(x y)v(y)dy J(x y)dyv(x) f (λu 1 + (1 λ)u 2 )v + lw, in (, T ), for some λ(x, t) [, 1]. We also have initial and boundary conditions (3.54) v(, x) = v (x), w(, x) = w (x), w (3.55) n =. Multiplying equation (3.52) by v t, integrating over, multiplying equation (3.52) by v, integrating over, multiplying equation (3.53) by w, integrating over, we have (3.56) (3.57) (3.58) v t 2 = v t v = J(x y)v(y)dyv t dx J(x y)dyv(x)v t (f (λu 1 + (1 λ)u 2 )vv t + lwv t )dx, J(x y)v(y)dyvdx J(x y)dyv 2 (f (λu 1 + (1 λ)u 2 )v 2 + lwv)dx, (w t w + lv t w) = w 2 dx, Adding equations (3.56)-(3.58) together, using Holder s and Young s inequalities, we have d (3.59) [w 2 + v 2 ]dx C 2 (T, R) [w 2 + v 2 ]dx dt for some positive constant C 2 (T, R). Inequality (3.59) and Gronwall s inequality imply the uniqueness and continuous dependence on initial data of the solution of (3.6)-(3.7). Denote Q T = (, T ), we have the following theorem: Theorem 3.2. If assumptions (P 1 ), (P 2 ) are satisfied, u L () and θ L H 1 (), then there exists a unique solution (u, θ) C([, T ], L ()) to the system (3.6)-(3.9) such that u t L (Q T ), and u tt, θ t, θ L 2 (Q T ).

37 Results concerning the asymptotic behavior of solutions follow along similar, though somewhat more complicated, lines as for the nonlocal Cahn- Hilliard equation above. Here the results are summarized without proof. Recall that I (θ + lu ) is conserved. Theorem 3.3. There exists a constant C(I ) otherwise independent of initial data such that u r C(I ), θ H 1 C(I ) for t t (I ). For the following results let X = {φ : Aφ φ φ L p, ν φ = } and let X α be the space D(A α ) endowed with the graph norms. α of A α for n 2p < α < 1. Theorem 3.4. Suppose that conditions (P 1 ) and (P 2 ) are satisfied and (u, θ ) L X α. Then the solution (u, θ) L X α satisfies sup θ(t) X α C 1 ( θ X α, u ) t< sup u(t) C 2 ( θ X α, u ) t< lim θ θ W 1,q = lim u t 2 =, t t for some q > n. If u (x) W 1,σ (), θ (x) W 2,σ () for σ > n, and f + J(x y)dy C >, then there exists a subsequence t k such that (u(t k ), θ(t k )) (u, c) in C γ (), where (u, c) is a steady state solution Pulses for the Nonlocal Wave Equation The results here are with Chunlei Zhang and are given in full in [2]. We consider the nonlocal wave equation for u(x, t): (4.1) u tt 1 ε 2 (j ε u u) + f(u) =, for t > and x R, where ε is a positive parameter and the kernel j ε of the convolution is defined by j ε (x) = 1 ε j(x ε ), where j( ) is an even function with unit integral. We assume that f is a C 2 function, satisfying f() = and f(ζ ) >, where ζ = inf{ζ > : F (ζ) = } and F (ζ) = ζ f(s)ds.

38 38 PETER W. BATES Typical examples include the quadratic function f(u) = u(u a) or the cubic f(u) = u(u b)(u + c) with a, b, c >. (4.2) We also consider a lattice version ü n 1 ε 2 k= α k u n k + f(u n ) =, n Z. 1 Note that, as ε, ε 2 (j ε u u) du xx, formally and in some weak sense described in [1], where d is a constant determined by j. So we can also regard (4.1) as a nonlocal version of the standard nonlinear wave equation (4.3) u tt du xx + f(u) =. In this paper we will study homoclinic traveling wave solutions of (4.1), i.e., solutions of the form u(x, t) = u(x ct) which decay at infinity. It is worth mentioning here that the parabolic versions (4.4) u t 1 ε 2 (j ε u u) + f(u) =, for t > and x R, and (4.5) u n 1 ε 2 k= α k u n k + f(u n ) =, n Z, where f is bistable (e.g., the cubic above) were treated in [1] and [9], respectively, where traveling or stationary waves were shown to exist, connecting the stable zeros of f. Certain assumptions are needed upon j and the α k s but we note that they are not required to be non-negative, i.e., they may change sign. When the wave has nonzero velocity, then the results in those papers are perturbative and rely upon spectral theory that we develop for the linearized operators for ε > sufficiently small. When the wave is stationary (the potential has wells of equal depth), then under the conditions imposed on the coefficients, solutions exist for all ε >. In this paper, we make slightly different assumptions on j and the α k s than in [1] and [9], and the proofs are very different, but in some sense spectral analysis is still involved. To be more precise, we assume (W 1 ) f C 2 (R), f() =, f () = a < ; f(ζ ) >, where ζ inf{ζ > : F (ζ) = } and F (ζ) = (W 2 ) j(x) L 1 (R) is even, has unit integral, ζ f(s)ds. ĵ(z) 1 lim z z 2 = d and ĵ(z) 1 d 1 z 2,