THE DEGIORGI-NASH-MOSER TYPE OF ESTIMATE FOR PARABOLIC VOLTERRA INTEGRODIFFERENTIAL EQUATIONS. Bei Hu and Hong-Ming Yin

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1 pacific journal of mathematics Vol. 178, No., 1997 THE DEGIOGI-NASH-MOSE TYPE OF ESTIMATE FO PAABOLIC VOLTEA INTEGODIFFEENTIAL EQUATIONS Bei Hu and Hong-Ming Yin The DeGiorgi-Nash-Moser estimate plays a crucial role in the study of quasilinear elliptic and parabolic equations. In the present paper we shall show that this fundamental estimate holds for solutions of a linear parabolic Volterra integrodifferential equation: u t = x i a ij (x, t) u x j ] t + x i b ij (x, t, τ) u ] dτ, x j where {a ij } and {b ij } are only assumed to be measurable, bounded and {a ij } satisfy a strong ellipticity condition. The proof is based on L,µ theory for parabolic equations. A global solvability result in the classical sense for a class of quasilinear parabolic integrodifferential equations is presented as an application of the general results. 1. Introduction. In this paper we consider the following linear parabolic Volterra integrodifferential equation (PVIDE): u t = a ij (x, t) u ] t + b ij (x, t, τ) u ] (1.1) dτ, (x, t) Q T, x i x j x i x j where Q T =Ω (,T],Ω is a bounded domain in n with boundary S = Ω C 1. The equation arises from a variety of mathematical models in engineering and physical sciences. For example, in the study of heat conduction in materials with memory, the classical Fourier s law is replaced by the following form (cf. 1]): t (1.) heat flux q = d u k(x, t, τ)u(x, τ)]dτ, where u is the temperature, d the diffusion coefficient and the integral term represents the memory effect in the material. The conservation of energy 65

2 66 BEI HU AND HONG-MING YIN implies that u(x, t) will satisfy Eq. (1.1) with an inhomogeneous term, provided that the temperature is assumed to be known for t. Another example is from a diffusion process in a glassy polymer (cf. ], 7] and the references therein). Experiments indicate that the classical Fickian law does not match the diffusion process. A non-fickian law (1.) is used to model the problem, where u(x, t) represents the concentration of the glassy polymer. Again the conservation of mass leads to a same type of equation as (1.1). The study for this type of equations has drawn considerable attention (cf. 1], 4]-7], 16]-17], etc.). The reader can find many more references in 1] and 13]. From a mathematical point of view, one would expect that the integral term should be dominated by the leading term in the Equation (1.1). Therefore, the theory of parabolic equations should be applicable to this type of equations. This is indeed true in some cases. Previous research strongly suggests that a solution of PVIDE has many similar properties to those of a parabolic equation. In 16] we have shown that the global Schauder estimate and a Wp,1 -estimate hold. More recently, the author of 17] considered a general equation u t = div A(x, t t, u, u x )+a(x, t, u, u x )+ div B(x, t, τ, u, u x )dτ. Under some structure conditions on A, B and a similar to the case of parabolic equations, the existence of a unique weak solution is established. egularity of the weak solution is also investigated. On the other hand, there are many essential differences between a PVIDE and a parabolic equation. For instance, the solution to a parabolic Volterra integrodifferential equation does not obey the maximum principle. A counterexample can be constructed without much difficulty. Therefore, it is a challenge to deduce an a priori L -bound of a solution. Unfortunately, it seems difficult to use the energy method to derive such a bound (see 17] for some special cases). Another essential difference concerns the regularity of solutions. By the DeGiorgi- Nash-Moser theory, we know that the solution of a parabolic equation with measurable coefficients {a ij } is Hölder continuous in the interior of the domain. No information is needed from initial and boundary data, as long as a weak solution exists. However, because of the nonlocal integral term in parabolic Volterra integrodifferential equations, the regularity of a solution strongly depends on the regularity of initial value. The DeGiorgi-Nash-Moser s estimate is a milestone in the study of quasilinear elliptic and parabolic equations. In the present paper, we shall derive this fundamental a priori estimate for weak solutions of a PVIDE. The argument of the proof is based on various estimates in Campanato space

3 DEGEOGI-NASH-MOSE ESTIMATE 67 L,µ (Q T ), where <µ<n+, which were developed recently for parabolic equations (cf. 18]). This L,µ type estimate gives the precise dependency of Hölder norm of a solution, without a priori assuming u(x, t) to be bounded. This theory enables us to use various integral estimates to replace the classical norm of a Hölder space. This method is widely used in the study of regularity of solutions of elliptic equations and systems (cf. 8], 11], 14], etc.). It turns out that this theory for parabolic equations is also very powerful. This paper is prepared in the following manner. We will state the main results in Section and present the proofs in Section 3. In Section 4, we employ the preceding results to obtain a global solvability result for a class of quasilinear PVIDE s in the classical sense.. Notation and Main esults. We shall introduce some standard notation for reader s convenience: Apoint(x, t) inq T will be denoted by z. The distance between two points z 1 =(x 1,t 1 ) and z =(x,t ) is equal to { } max x 1 x, t 1 t 1. For r>, B r (x )={x n : x x <r} and Q r (z )=B r (x ) t r,t ], where z =(x,t ). Let 1 u z,r = udz, Q r (z ) Q r(z ) where Q r (z ) denotes the Lesbegue measure of Q r (z )in n+1. For µ> and z =(x,t ), let u],µ,qr = ( sup ρ µ u u z,ρ dz z Q r, <ρ<r Q ρ(z ) We shall use Q r QT in the integration whenever Q r is not a subset of Q T. The space L,µ (Q r ) consists of all functions in L (Q r ) such that u],µ,qr <. L,µ (Q r ) is a Banach space with the norm u,µ,qr = { u L (Q r) +u],µ,q r } 1. ) 1.

4 68 BEI HU AND HONG-MING YIN The Banach spaces H 1 (Ω),L (,T;H 1 (Ω)),C α, α (QT ), etc. are the same as those in 1]. epeated subscript means a summation. To the Equation (1.1), we append the following initial and boundary conditions: (.1) (.) u(x, t) =g(x, t) on S T = Ω,T], u(x, ) = u (x) on Ω. Throughout this paper, the strong ellipticity condition is assumed regarding the measurable coefficients a ij : (.3) a ij = a ji, λ ξ a ij ξ i ξ j Λ ξ for ξ n, where <λ Λ < are constants. emark 1. Note that no ellipticity assumption is required regarding the coefficients {b ij }. Definition. A function u L (,T;H 1 (Ω)) is said to be a weak solution of (1.1) in Q T satisfying (.1)-(.), ifu g L (,T;H(Ω)) 1 (here we assume that g(x, t) can be extended to Q T ) and the following integral identity holds: Q T t ] ] uϕ t + a ij u xj + b ij u xj dτ ϕ xi dxdt = u ϕ(x, )dx Ω for any test function ϕ C 1 (Q T ) with ϕ(x, t) =ont=t and S T. The existence of a unique weak solution of (1.1), (.1)-(.) can be proved by means of the finite element method (cf. 17]). Our objective is to obtain a priori estimates of the DeGiorgi-Nash-Moser type which imply the Hölder continuity of the weak solution u(x, t) of(1.1), (.1)-(.). We begin by obtaining an interior estimate : Theorem.1. Let T>and Q Q T with d = dist{q, S T } >. Let a ij (x, t), b ij (x, t, τ) be measurable and {a ij } satisfy (.3). Assume that n ] aij L (Q T ) + b ij L (Q T ) M<, i,j=1 and that u <, L,(µ ) + (Ω) where µ = n +β for some β (, 1). Then there exist constants α (, 1) and C>such that any L (,T;H 1 (Ω)) solution u(x, t) of (1.1), (.1)-(.) on Q T satisfies u C α, α C, (Q)

5 DEGEOGI-NASH-MOSE ESTIMATE 69 where the constant C and the Hölder exponent α depend only on d, M, λ, T, u L (,T ;H 1 (Ω)) and u L,(µ ) + (Ω). emark. As we mentioned in Introduction, the interior estimate in Theorem.1 holds actually down to the bottom t =, but away from the lateral boundary S T. This interior estimate is not valid without the required regularity on u. Next, we obtain a DeGiorgi-Nash-Moser type estimate valid up to the boundary: Theorem.. In addition to assumptions of Theorem.1, assume that there exists a function ψ(x, t) H 1 (,T;H 1 (Ω)) such that ψ(x, t) =g(x, t) on S T, and ψ(x, ) = u (x) on Ω in the sense of trace. If ψ t,ψ xi L,µ (Q T ) for i =1,,n, then there exist constants C and α (, 1) such that (.4) u C α, α (Q T ) C, where the constants C and α depend only on ψ t L,µ (QT ) + ψ L,µ (QT ) and the data in Theorem.1. emark 3. The estimate (.4) is also true for the following conormal boundary condition: a ij u xj cos(n i,x i )=g(x, t), where n is the outward normal on S, provided that g(x, t) is uniformly bounded. emark 4. The equation is linear for u. By renormalization of the equation, one can write the dependence on the initial data in the interior estimate as follows. u C α, α C ( u L (,T ;H 1 (Ω)) + u L,µ (Ω)). A similar remark also applies to the global estimate. emark 5. We stated our Theorems.1 and. for the operator containing only the highest order derivative. All the results can be extended without any difficulty to the following operator Lu] = u t { x i a ij (x, t) u ] + b i (x, t) u + cu x j x i

6 7 BEI HU AND HONG-MING YIN + t ( x i b ij (x, t, τ) u ] + d i (x, t, τ) u ) } (x, τ)+e(x, t, τ)u(x, τ) dτ x j x i + j f j x j + f, provided that suitable conditions on the coefficients of lower order terms hold. 3. Proofs. We shall first state the following interior DeGiorgi-Nash-Moser type estimate. Let x Ω be arbitrary with d = dist{x,ω} >. Let Q =, T B (x ) ( T, T ], where T,T]. We shall restrict d. Lemma 3.1. Let v(x, t) be a solution to the following equation (no initialboundary conditions needed) v t = a ij (x, t) v ] in Q x j x,, T i then there exists constants C 1 > and α (, 1) such that for any r (,] v dxdt C 1 ( r ) n+α v dxdt, Q r, T Q, T where the constants C 1 and α depend only on λ and Λ, they are independent of r and. This is the same as Lemma.4 in 18]. We shall not repeat the proof. For the convenience of the proof we extend the solution u(x, t) tot< by letting u(x, t) u (x) for t<. We also define a ij (x, t) λδ ij for t<. Our key estimate is the next lemma. As indicated in the introduction, the estimates will have to depend on the initial data, owing to the nonlocal nature of the equation. Lemma 3.. Let u(x, t) L (,T;H 1 (Ω)) be a weak solution of Equation (1.1). Suppose that u H 1 (Ω) such that for any x Ω B (x ) u dx M 1 n +β for any << d,

7 DEGEOGI-NASH-MOSE ESTIMATE 71 (without loss of generality, we may assume that β < α, where α is the constant in Lemma 3.1). (1). (Interior estimates) If dist { Q,S, T T} >, then Q, T u dxdt C n+β, where the constant C depends only on λ, Λ,d, the constant M in Theorem.1, the constant M 1, and the L (,T;H 1 (Ω))-norm of the solution u. (). (Boundary estimates) Moreover, if u satisfies the boundary condition as specified in Theorem., then for any T >, u dxdt C n+β, Q (Ω 1, T ]), T where the constant C depends only on the boundary data in addition to the quantities in the interior estimate (1) above. Proof. Let v(x, t) be the solution of the following problem: v t = a ij (x, t) v ] x j x i in Q,, T v(x, t) =u(x, t) on p Q., T It is then clear that u v satisfies the following equation in the weak sense (u v) t = ] (u v) (3.1) a ij + b j (x, t)+ f j (x, t), x j x i x j x j where t u(x, τ) b ij (x, t, τ) dτ for t (3.) b j (x, t) = x i for t<, for t f j (x, t) = λ u (x) for t<. x j Multiplying this equation with u v and integrating over Q, T, we obtain sup (u v) dx + λ (3.3) (u v) dxdt B T τ T 1 λ j Q, T (b j + f j ) dxdt. Q, T

8 7 BEI HU AND HONG-MING YIN Observe that the estimate in Lemma 3.1 holds for v independently of the initial and boundary conditions on the parabolic boundary p Q. Indeed,, T for any r we have from Lemma 3.1 Q r, T v dxdt C 1 ( r ) n+α Q, T v dxdt. When r, the estimate is trivial. Using Lemma 3.1 and (3.3), we now get, for any r, u dxdt Q r, T Q r, T C 1 ( r v dxdt + (u v) dxdt ) n+α Q r, T v dxdt + (u v) dxdt Q, T Q, T (3.4) ( ) r n+α 4C 1 Q, T ( ) r n+α + 4C 1 + u dxdt ] (u v) dxdt 4C 1 ( r ) n+α Q, T u dxdt +(4C 1 +) (u v) dxdt 4C 1 ( r ) n+α Q, T Q, T u dxdt + (C 1 +1) λ Q, T j Q, T b j + f j dxdt. We now estimate b j (x, t). ecalling that M is the bound for the coefficients b ij (x, t), we have (by Hölder s inequality) (3.5) b j (x, t) M T t u x (x, τ)dτ M T i T u x (x, τ)dτ. i It follows that b j (x, t) dxdt Q, T M T T B u dxdt

9 DEGEOGI-NASH-MOSE ESTIMATE 73 (3.6) M T T/ ] T k u dxdt k= T (k+1) B M T ( T/ ] ) T k +1 max u dxdt k T/ ] T (k+1) B M T ( T + ) T k max u dxdt, k T/ ] T (k+1) B where T/ ] is the integer part of T/. By assumption, we have (3.7) f j (x, t) dxdt C n+β. Q, T Now let T > to be specified later and x Ω such that B (x ) Ω. Set (notice that u(x, t) has already been extended to t<) (3.8) g(, x ) = t sup u dxdt. t T t B (x ) Then by (3.4), (3.6) and (3.7), ( ) r n+α g(r, x ) 4C 1 + 4(C ] 1 +1) (3.9) M T (T + ) g(, x λ ) + C n+β. Note that we have assumed that β<α. By 8] (Lemma.1, p. 86), there exists ε >, depending only on α, β and C 1 (ε is independent of C ), such that if (4C 1 +) (3.1) M T (T + ) ε λ, then (3.11) g(r, x ) C ( ) ] r n+β g(, x )+C r n+β, where the constant C depends only on α, β and C 1. The condition (3.1) is satisfied if we choose T to be suitably small. By a well-known iteration process (cf. 8]), we have (3.1) u dxdt C n+β if T T,B Ω, Q, T

10 74 BEI HU AND HONG-MING YIN where C depends only on L (,T;H 1 (Ω))-norm of u(x, t), β and the constant in (3.11). Notice that the estimate (3.1) is good for small T satisfying (3.1). Next we want to show that this estimate can be extended step-by-step in t direction with exactly the same step length (equal to T ) in each step. Therefore the estimate is valid for any given finite time interval. Suppose that the estimate (3.1) is correct for < T kt. We rewrite the equation as (3.13) u t = x j a ij (x, t) u ] ( t + + x i kt kt ) x i b ij (x, t, τ) u ] dτ. x j All the previous estimates are still valid, with (3.14) f j (x, t) = i kt b ij (x, t, τ) u(x, τ) dτ. x i By the induction hypothesis we have, for kt < T <(k+1)t, (3.15) j Q, T kt f j (x, t) dxdt M kt u dxdτ B ( kt M kt ] ) +1 n+β C(k) n+β, where the constant C(k) may increase with each step. Now the exact procedure as in (3.1) to(3.1) implies that ( ) ] r n+α (3.16) g(r, x ) 4C 1 + ε g(, x )+C C(k) n+β. Since the choice of ε in (3.1) is independent of the constant C C(k), the estimate (3.1) extends to kt < T < (k +1)T. This proves the interior estimates. From the proof, we see that the estimates (3.1) extends to the lateral boundary if Lemma 3.1 extends to the lateral boundary. We know that such results are valid (see 11]) if either a Dirichlet or Neumann boundary condition is imposed on the lateral boundary and the boundary value satisfies the condition stated in Theorem.. We shall skip the detail here. Finally, let us finish the Proof of Theorems.1 and.. In order to show that u(x, t) is Hölder continuous, we shall employ the parabolic embedding

11 DEGEOGI-NASH-MOSE ESTIMATE 75 theorem (Lemma.6 in 18]). By applying Lemma.6 in 18], we have for any Q ρ Q T with dist{b ρ,ω}>, (u u z,ρ) dz Q ρ Cρ u dz + Cρ Q ρ Cρ Q ρ u dz + ρ Cρ n++β, B ρ n Q ρ j=1 t ] f j + b j dz u dτdz + ρ B ρ u dx where at the final step we have used the estimates for f j and b j. As L,n++β (Q) for any β (, 1) is equivalent to C β, β (Q) (cf. PropositionAin14]) it follows that u(x, t) ishölder continuous in Q r = B r (x ),T]. As r is independent of solution and x is arbitrary in Ω, by a finite covering technique we conclude that u(x, t) is Hölder continuous in Q as long as dist{q, S T } >. This completes our proof for Theorem.1. Using the second part of Lemma 3. and the same argument as above, we can establish Theorem.. ] 4. Applications. In this Section, we shall use the results obtained in the preceding section to show global existence of a classical solution for nonlinear parabolic Volterra integrodifferential equations. Consider the following problem: (4.1) u t = x i a ij (x, t, u) u ] t + x j (4.) u(x, t) =g(x, t) on S T, (4.3) u(x, ) = u (x) on Ω. x i b ij (x, t, τ, u) u ] dτ in Q T, x j Theorem 4.1. Assume that the coefficients a ij (x, t, s) satisfy the ellipticity ( ) ( condition (.3), a ij, x k a ij ), (a u ij C α, α,β (Q T 1 ), and b ij, x k b ij ), ) (b u ij C α, α, α,β (Q T,T] 1 ), where α, β (, 1). Moreover, there exists a function ψ(x, t) C +γ,1+ γ (QT )(γ=αβ) such that ψ(x, t) =g(x, t)

12 76 BEI HU AND HONG-MING YIN on S T and ψ(x, ) = u (x) on Ω. Then the problem (4.1)-(4.3) exists a unique classical solution. Proof. Local existence of a classical solution is standard from the theory of parabolic equations and contraction mapping principle. By using the method of continuity, we know that global existence relies on an a priori estimate in the space C +α,1+ α (QT ). Multiplying the Equation (4.1) byu g(x, t), we can easily obtain From Theorem., we know u L (,T ;H 1 (Ω)) C. u C α, α (Q T ) C. If we rewrite the Equation (4.1) in nondivergence form, then the coefficients are Hölder continuous. Moreover, their Hölder norms are bounded by known data. Using the global Schauder estimate for parabolic Volterra integrodifferential equations (cf. 16]), we obtain u C +γ,1+ γ (Q T ) C (γ = αβ), where C depends only on known data. Consequently, we have the desired result. Acknowledgment. helpful comments. The authors thank Professor G. Lieberman for his eferences 1] G. Andrews, On existence of solutions to the equation u tt = u xxt + σ(u x) x, J. Diff. Eqs., 35 (198), -31. ] D.S. Cohen and A.B. White, Sharp fronts due to diffusion and viscoelastic relaxation in polymers, SIAM J. Appl. Math., 51 (1991), ] S. Campanato, Equazioni paraboliche del secondo ordine e spazi L,θ (Ω,δ), Annali di Matem. Pura e Appl., 73 (1966), ] M.G. Crandall, S.O. Londen and J.A. Nohel, An abstract nonlinear Volterra integrodifferential equation, J. Math. Anal. Appl., 64 (1978), ] C.M. Dafermos, The mixed initial boundary value problem for the equations of nonlinear one dimensional visco-elasticity, J. Diff. Eqs., 6 (1969), ] G. Da Prato and M. Iannelli, Existence and regularity for a class of integrodifferential equations of parabolic type, J. Math. Anal. Appl., 11 (1985),

13 DEGEOGI-NASH-MOSE ESTIMATE 77 7] A. Friedman, Mathematics in Industrial Problems, Springer-Verlag, New York, ] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton University Press, New Jersey, ] J. Moser, On a pointwise estimate for parabolic differential equations, Comm. on Pure and Applied Mathematics, 4 (1971), ] J.A. Nohel, Nonlinear Volterra equations for heat flow in materials with memory, Integral and functional differential equations, Lecture notes in pure and applied mathematics, edited by T.L. Herdman, S.M. ankin, III, H.W. Stech, Marcel Dekker Inc., ] G.M. Troianiello, Elliptic Differential Equations and Obstacle Problems, Plenum Press, New York, ] O.A. Ladyzenskaja, V.A. Solonnikov and N.N. Ural ceva, Linear and Quasilinear Equations of Parabolic type, AMS translation Monograph, Vol. 3, Providence, hode Island, ] M. enardy, W.J. Hrusa and J.A. Nohel, Mathematical Problems in Viscoelasticity, Longman Scientific and Technical, New York, ] M. Struwe, On the Hölder continuity of bounded weak solutions of quasilinear parabolic systems, Manscripta Mathematica, 35 (1981), ] H.M. Yin, The classical solutions for nonlinear integrodifferential equations, J. Integral Equations and Applications, 1 (1988), ], On parabolic Volterra equations in several space dimensions, SIAM J. on Mathematical Analysis, (1991), ], Weak and classical solutions of some Volterra integrodifferential equations, Communication in P.D.E., 17 (199), ], L,µ theory for parabolic equations and applications, J. of Partial Differential Equations, 1(1) (1997). eceived June 15, 1995 and revised November 1, The work of the first author is partially supported by NSF grant DMS University of Notre Dame Notre Dame, IN address: Bei.Hu.1@nd.edu or Hong-Ming.Yin.3@nd.edu