On some nonlinear parabolic equation involving variable exponents

On some nonlinear parabolic equation involving variable exponents Goro Akagi (Kobe University, Japan) Based on a joint work with Giulio Schimperna (Pavia Univ., Italy) Workshop DIMO-2013 Diffuse Interface Models Grand Hotel Bellavista, Levico Terme, Italy Sep. 10 13, 2013

Doubly nonlinear parabolic equation Nonlinear parabolic equations of the form β( t u) u = f in (0, T ) with a maximal monotone graph β : R R, a domain of R N, and a given function f = f(x, t) : (0, T ) R have been studied in various contexts. The linear Laplacian is often replaced with nonlinear variants such as the so-called m-laplacian m given by m u = div ( u m 2 u ), 1 < m <, and then, the equation above is called a doubly nonlinear parabolic equation. 1/36

Doubly nonlinear evolution equation By setting u(t) := u(, t), such a nonlinear parabolic equation is interpreted as an abstract evolution equation, A(u (t)) + B(u(t)) = f(t) in X, 0 < t < T, ( ) with unknown function u : (0, T ) X, two (possibly nonlinear) operators A, B in X and f : (0, T ) X. It is quite natural to build the existence and regularity theory for ( ) in some class of vector-valued functions, like the Lebesgue-Bochner space L p (0, T ; X). 2/36

Abstract theories A(u (t)) + B(u(t)) = f(t) in X, 0 < t < T. Barbu ( 75), Arai ( 79), Senba ( 86) established in L 2 (0, T ; H); time differentiation of the equation, which transforms it into another (more tractable) type of doubly nonlinear equation; some peculiar monotonicity condition, (Bu Bv, A(u v)) H 0. 3/36

Abstract theories (contd.) A(u (t)) + B(u(t)) = f(t) in X, 0 < t < T. Colli-Visintin ( 90), Colli ( 94) established in L 2 (0, T ; H) or L p (0, T ; V ), 1 < p < ; a power growth for A : V V, c 0 u p V Au, u V + C, Au p V C( u p V + 1). These approaches have been extensively studied by many authors. Recently, it was extended by Mielke-Rossi-Savaré ( 13) to a quite broad class of doubly nonlinear evolution. 4/36

Doubly nonlinear parabolic eq. with power nonlinearity Let us consider t u p 2 t u m u = f(x, t) in Q := (0, T ), where 1 < p, m <. u = 0 on (0, T ), u(, 0) = u 0 in, It is a simple case of the generalized Ginzburg-Landau eq. (Gurtin 96), β(u, u, t u) t u = div [ u ψ(u, u)] u ψ(u, u) + γ. This problem falls within the framework of Colli ( 94) in L p (0, T ; V ) by setting V = L p (), Au = u p 2 u, Bu = m u. Indeed, the p-power growth condition holds: u p V = Au, u V, Au p V = u p V. 5/36

Target problem of this talk Let R N be a smooth bounded domain. In this talk, we discuss (1) (2) (3) t u p(x) 2 t u m(x) u = f(x, t) in Q := (0, T ), u = 0 on (0, T ), u(, 0) = u 0 in, where 1 < p(x), m(x) < are variable exponents and m(x) stands for the m(x)-laplacian given by m(x) u = div ( u m(x) 2 u ). Eq. (1) can describe mixed settings of several types of (generalized) Ginzburg-Landau models, e.g., p(x) 2 in }{{ 1, } p(x) 2 in 2, }{{} = 1 2. usual GL model generalized GL model 6/36

Lebesgue and Sobolev spaces with variable exponents p := ess inf x p(x) and p+ := ess sup x P() := { p M() : 1 p }, p(x), P log () := {p P() : log-hölder continuous}. Variable exponent Lebesgue and Sobolev spaces are defined as follows: { } L p(x) () := u M() : u(x) p(x) dx <, u L p(x) () := inf { λ > 0 : u(x) λ p(x) dx 1 W 1,p(x) () := { u L p(x) () : xi u L p(x) () for i = 1,..., N }, ( 1/2 u W 1,p(x) () := u 2 L p(x) () + u 2 L ()). p(x) }, 7/36

Difficulties arising from variable exponents t u p(x) 2 t u m(x) u = f(x, t) Following the strategy for the constant exponent case, one may set { } V = L p(x) () = u M() : u(x) p(x) dx < and note that ( u p(x) 2 u ) u = u p(x), u p(x) 2 u p (x) = u p(x). However, it only implies, e.g., u p(x) 2 u, u V c u p V, u p(x) 2 u (p+ ) C u p+ V V with p + := ess sup p(x), p := ess inf p(x) and V = L p (x) (). Since p + > p, it does not fall within the frame of L p (0, T ; V ). 8/36

Difficulties arising from variable exponents (contd.) More precisely, testing the following equation by t u, t u p(x) 2 t u m(x) u = 0, we have t u p(x) dx + u m(x) 2 u ( t u) dx = 0. }{{} t ( 1 m(x) u m(x) ) The integration of both sides over (0, t) implies t 1 τ u p(x) dxdτ + 0 m(x) u(x, t) m(x) dx 1 m(x) u(x, 0) m(x) dx. 9/36

Difficulties arising from variable exponents (contd.) To estimate t u p(x) 2 t u in L p (x) () = (L p(x) ()), we use t u p(x) 2 t u p (x) dx = t u p(x) dx. The Lebesgue-Bochner space L q (0, T ; V ) approach: One needs to estimate t u and t u p(x) 2 t u in some L-B spaces. However, in the variable exponent setting, we should emphasize that w(x) p(x) dx w p(x) for w L p(x) (). L p(x) () Instead, we use σ p(x) ( w L p(x)) w(x) p(x) dx σ + p(x) ( w L p(x)) w Lp(x) () with σ p(x) (s) := min{sp, s p+ } and σ + p(x) (s) := max{sp, s p+ }. 10/36

Difficulties arising from variable exponents (contd.) Then for V = L p(x) (), one may obtain estimates, e.g., T T ( ) t u p V dt t u p(x) dx dt, T 0 0 t u p(x) 2 t u (p ) V 0 dt T 0 ( ) t u p(x) dx However, since (p ) = (p + ) < (p ) by p + > p, we note that ( L (p ) (0, T ; V ) L p (0, T ; V )). This fact will prevent us to identify the limit of approximate solutions to prove the existence of solution. dt. 11/36

Lebesgue space approach Recall the estimate t u p(x) 2 t u p (x) dx = Then we obtain t u p(x) 2 t u p (x) dxdt = Q with Q := (0, T ), which implies, e.g., t u p(x) dx. Q t u p(x) dxdt t u p(x) 2 t u ( p L p (x) (Q) t u L (Q)) + /p. p(x) Here we note that L p (x) (Q) ( L p(x) (Q) ). 12/36

Lebesgue space approach (contd.) Namely, setting V := L p(x) (Q) ( ) V = L p (x) (Q), Q = (0, T ) and defining A : u u p(x) 2 u for u V, we observe that A : V V is bounded and coercive. In order to treat the equation in L p(x) (Q) L p (x) (Q), we note that: No loss of integrability (in t); L p(x) (Q) L p(x) (0, T ; L p(x) ()), which is meaningless for p(x); lack of devices: e.g., chain-rules for subdifferential are usually formulated in Lebesgue-Bochner space settings. 13/36

Aim of this talk We shall discuss the existence and regularity of solutions for t u p(x) 2 t u m(x) u = f(x, t) in Q := (0, T ), u = 0 on (0, T ), u(, 0) = u 0 in. To this end, we shall work in a mixed frame of Lebesgue space L p(x) (Q) and usual Lebesgue-Bochner spaces; develop devices of subdifferential calculus, in particular, a chain-rule for subdifferentials in the mixed frame. 14/36

Basic assumptions (H) (H1) (H2) (H3) m P log (), p P(), 1 < p, m, p +, m + <, ess inf x (m (x) p(x)) > 0, m (x) := f L p (x) (Q), Nm(x) (N m(x)) +, u 0 W 1,m(x) 0 (). Remark. By (H1), L p(x) () and W 1,m(x) () are uniformly convex and separable Banach spaces. Since m( ) P log (), one can define W 1,m(x) 0 () := C 0 ()W 1,m(x) (), u W 1,m(x) 0 () := u L m(x) (). Moreover, (H2) ensures that W 1,m(x) 0 () compact L p(x) (). 15/36

Main results Definition 1 (Strong solutions) We call u L p(x) (Q) a strong solution of (1) (3) in Q whenever the following conditions hold true: (i) t u(, t) is continuous with values in L p(x) () on [0, T ], and it is weakly continuous with values in W 1,m(x) 0 () on [0, T ], (ii) t u L p(x) (Q), m(x) u L p (x) (Q), (iii) the equation (1) holds for a.e. (x, t) Q, (iv) the initial condition (3) is satisfied for a.e. x. Theorem 2 (Existence) Assume (H). Then the Cauchy-Dirichlet problem (1) (3) admits (at least) one strong solution u. 16/36

Main results (contd.) Theorem 3 (Time-Regularization of strong solutions) In addition to (H), suppose that t t f L p (x) (Q). Then, the Cauchy-Dirichlet problem (1) (3) admits a strong solution u, which additionally satisfies ess sup t (δ,t ) t u(, t) L p(x) () <, ess sup t (δ,t ) m(x) u(, t) L p (x) () < for any δ (0, T ). 17/36

Lebesgue-Bochner space setting Set V = L p(x) () and X = W 1,m(x) 0 () with u V := u L (), p(x) u X := u L (), m(x) v, u V = u(x)v(x) dx for all u V, v V = L p (x) (). By (H2), it follows that X compact V and V compact X. Define functionals ψ and φ on V by 1 ψ(u) := p(x) u(x) p(x) dx φ(u) := for u V 1 m(x) u(x) m(x) dx if u X, if u V \ X. 18/36

Lebesgue-Bochner space setting (contd.) Denote by ( Q ) the subdifferential in V = L p(x) () (V = L p(x) (Q)). Then (1) (3) can be reduced to ψ(u (t)) + φ(u(t)) = P f(t) in V, 0 < t < T, u(0) = u 0, where P f(t) := f(, t). 19/36

For each u M(Q), write Identification between two settings P u(t) := u(, t) for t (0, T ). Proposition 4 (Identification between L-B and L spaces) Let 1 p < and let p(x) be such that 1 p p + <. (i) P is a linear, bijective, isometric mapping from L p (Q) to L p (0, T ; L p ()). Furthermore, if u L p(x) (Q), then P u L p (0, T ; L p(x) ()). (ii) The inverse P 1 : L p (0, T ; L p ()) L p (Q) is well-defined, and for u = u(t) L p (0, T ; L p ()), u(t) = P 1 u(, t) for a.e. t (0, T ). Remark. L (0, T ; L ()) L (Q). 20/36

Identification between two settings (contd.) For each u M(Q), write P u(t) := u(, t) for t (0, T ). Proposition 4 (Identification between L-B and L spaces) Let 1 p < and let p(x) be such that 1 p p + <. (iii) If u L p(x) (Q) with t u L p(x) (Q), then P u belongs to the space W 1,p (0, T ; L p(x) ()) and (P u) = P ( t u). (iv) If u W 1,p (0, T ; L p ()), then t (P 1 u) L p (Q) and t (P 1 u) = P 1 (u ). 21/36

Identification of subdifferentials in two settings Let ϕ : V = L p(x) () (, ] be proper, l.s.c., convex and define Φ : V = L p(x) (Q) (, ] by T ϕ(p u(t)) dt if ϕ(p u( )) L 1 (0, T ), Φ(u) := 0 otherwise. Proposition 5 (Identification of subdifferentials) For u V, ξ V with 1 < p p + <, ξ Q Φ(u) iff P ξ(t) ϕ(p u(t)) for a.e. t (0, T ). 22/36

Lebesgue space setting Define Ψ and Φ on V = L p(x) (Q) as, for u V, Ψ(u) := Φ(u) := T 0 Q 1 p(x) u(x, t) p(x) dx dt = T 0 ψ(p u(t)) dt, φ(p u(t)) dt if P u(t) X for a.e. t (0, T ), t φ(p u(t)) L 1 (0, T ), otherwise. Then by the latter proposition, the evolution eqn ( (1) (3)) is rewritten by Q Ψ( t (P 1 u)) + Q Φ(P 1 u) = f in V, u(0) = u 0. 23/36

Construction of solutions Step 1 (time-discretization). For n = 0,..., N 1, ( ) un+1 u n ψ + φ(u n+1 ) = f n+1 in V, h where h := T /N, t n := nh and f n := 1 tn h t n 1 P f(θ) dθ. Moreover, define a piecewise constant interpolant u N : (0, T ) X = W 1,m(x) 0 (); a piecewise linear interpolant u N : (0, T ) X. Then we get ψ(u N (t)) + φ(u N (t)) = f N (t) in V, for a.e. t (0, T ) with u N (0) = u 0 in the Lebesgue-Bochner space setting, and equivalently, Q Ψ( t (P 1 u N )) + Q Φ(P 1 u N ) = P 1 f N in V, u N (0) = u 0 in the Lebesgue space setting. 24/36

Construction of solutions (contd.) Step 2 (energy estimates). Test discretized eqn. by (u n+1 u n )/h. Then t (P 1 u N ) p(x) dx dt + sup φ(u N (t)) C, t (0,T ] Q which also gives t (P 1 u N ) V C, sup t (0,T ] u N (t) X + sup t [0,T ] u N (t) X C. Recall that A = Q Ψ : v v p(x) 2 v is bounded from V to V. We conclude that Q Ψ( t (P 1 u N )) V C, which also implies the boundedness of Q Φ(P 1 u N ) in V by comparison. 25/36

Construction of solutions (contd.) Step 3 (Convergence). u N u strongly in C([0, T ]; V ), weakly star in L (0, T ; X), u N u strongly in L (0, T ; V ), weakly star in L (0, T ; X), P 1 u N û = P 1 u strongly in V, t (P 1 u N ) t û weakly in V, Q Φ(P 1 u N ) ξ weakly in V, Q Ψ( t (P 1 u N )) η weakly in V. Thus η + ξ = f in V. From the maximal monotonicity of Q Φ in V V, one can immediately obtain ξ Q Φ(û). It remains to show η Q Ψ( t û). 26/36

Construction of solutions (contd.) To this end, by a standard technique, one has Q Ψ ( t (P 1 u N ) ) t (P 1 u N ) dx dt = Q Q Q ( P 1 f N Q Φ(P 1 u N ) ) t (P 1 u N ) dx dt ( P 1 f N ) t (P 1 u N ) dx dt φ(u N (T )) + φ(u 0 ). Passing to the limit as N, we have lim sup Q Ψ ( t (P 1 u N ) ) t (P 1 u N ) dx dt n Q f t û dx dt φ(u(t )) + φ(u 0 ) =? η t û dxdt. Q Q 27/36

Construction of solutions (contd.) Here we need: Proposition 6 (Chain rule for subdifferentials in a mixed frame) Let p( ) P() satisfy 1 < p p + <. Let u V be such that t u V. Suppose that there exists ξ V such that ξ Q Φ(u). Then, the function t ϕ(p u(t)) is absolutely continuous over [0, T ]. Moreover, for each t (0, T ), we have d dt ϕ(p u(t)) = η, (P u) (t) V for all η ϕ(p u(t)), whenever P u and ϕ(p u( )) are differentiable at t. In particular, for 0 s < t T, we have ϕ(p u(t)) ϕ(p u(s)) = (s,t) ξ τ u dx dτ. 28/36

Construction of solutions (contd.) Applying the latter chain-rule, we conclude that lim sup Q Ψ ( t (P 1 u N ) ) t (P 1 u N ) dx dt n u=p û = Prop 6 = Q Q Q Q f t û dx dt φ(u(t )) + φ(u 0 ) f t û dx dt φ(p û(t )) + φ(p û(0)) f t û dxdt Q ξ t û dxdt ξ+η=f = Q η t û dxdt, whence follows η Q Ψ( t û). Consequently, û solves (1) (3). 29/36

Outline of proof for the chain-rule Step 1 (Approximation). We start with approximating the proper, l.s.c., convex functional ϕ defined on V = L p(x) () as follows: ( ) p(x) λ ϕ λ (u) := min v(x) u(x) v V p(x) λ dx + ϕ(v) which is a variant of Moreau-Yosida regularization of ϕ. for u V, Then ϕ λ enjoys similar properties to the usual M-Y regularization with the modified resolvent J λ and modified Yosida approximation A λ of A = ϕ. One can also define the modified Moreau-Yosida approximation Φ λ of Φ defined on V. 30/36

Outline of proof for the chain-rule Let A : V V be a maximal monotone operator. The modified resolvent J λ : V V of A is given by, for each u V, J λ u := u λ, which is a unique solution of ( ) uλ u Z + A(u λ ) 0 in V, λ where Z (u) := u p(x) 2 u for u V. The modified Yosida approximation A λ : V V of A is given by ( ) u Jλ u A λ (u) := Z A(J λ u) for each u V. λ 31/36

Outline of proof for the chain-rule Step 2 (Correspondence of ϕ λ and Φ λ ). Lemma 7 (Correspondence of ϕ λ and Φ λ ) It follows that Φ λ (u) = T 0 In particular, for u V and ξ V, ϕ λ (P u(t)) dt for all u V. ξ λ = Q Φ λ (u) iff P ξ λ (t) = ϕ λ (P u(t)) for a.a. t (0, T ). 32/36

Outline of proof for the chain-rule Step 3 (Chain-rule for ϕ λ ). Let u V be such that t u V. Then u, t u L p(x) (Q) P u W 1,p (0, T ; L p(x) ()). Moreover, since ϕ λ is bounded, we see that ϕ λ (P u( )) L (0, T ; V ). Let ξ λ := Q Φ λ (u) and use a standard chain-rule in the L-B space setting. ϕ λ (P u(t)) ϕ λ (P u(s)) chain = Lem 7 = = t s t s ϕ λ (P u(τ )), (P u) (τ ) V dτ P ξ λ (τ ), (P u) (τ ) V dτ (s,t) ξ λ t u dxdτ, 0 s t T. 33/36

Outline of proof for the chain-rule Step 4 (Boundedness and convergence). Lemma 8 (Boundedness of modified Yosida approx.) Let u V, η Au and let A λ be the modified Yosida approximation. 1 p (x) A 1 λu(x) p (x) dx (x) p (x) η(x) p dx. An analogous statement also holds for maximal monotone A : V V. Thus since ξ λ = Q Φ λ (P u), we have, for any η Q Φ(P u), 1 p (x) ξ 1 λ p (x) dxdt (x) p (x) η p dxdt <. Hence Q ξ λ ξ weakly in V and ξ Q Φ(P u). Q 34/36

Outline of proof for the chain-rule We have obtained ϕ λ (P u(t)) ϕ λ (P u(s)) = (s,t) ξ λ t u dxdτ. Using the fact ϕ λ (u) ϕ(u) u V, we have obtained the formula, ϕ(p u(t)) ϕ(p u(s)) = (s,t) ξ t u dxdτ, which also implies the absolute continuity of t ϕ(p u(t)). 35/36

Thank you for your attention! Goro Akagi Kobe University, Japan akagi@port.kobe-u.ac.jp http://www2.kobe-u.ac.jp/~akagi56/