Existence,uniqueness, and nonexistence of solutions to nonlinear diffusion equations with p ( x , t ) -Laplacian operator

Boundary Value Problems, Aug 2016

The aim of this paper is to deal with the existence and nonexistence of weak solutions to the initial and boundary value problem for u t = div ( | ∇ u | p ( x , t ) − 2 ∇ u + b ( x , t ) ∇ u ) + f ( u ) . By constructing suitable function spaces and applying the method of Galerkin’s approximation as well as weak convergence techniques, the authors prove the existence of local solutions. Furthermore, we choose a suitable test-function, make integral estimates, and apply Gronwall’s inequality to prove the uniqueness of weak solutions. At the end of this paper, the authors construct a suitable energy functional, obtain a new energy inequality, and apply a convex method to prove the nonexistence of solutions. Especially, it is worth pointing out that the results are obtained with the assumption that p t ( x , t ) is only negative and integrable, which is weaker than most of the other papers required.

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Existence,uniqueness, and nonexistence of solutions to nonlinear diffusion equations with p ( x , t ) -Laplacian operator

Gao et al. Boundary Value Problems Existence,uniqueness, and nonexistence of solutions to nonlinear diffusion equations with p(x, t )-Laplacian operator Yanchao Gao 0 Ying Chu 0 Wenjie Gao 1 0 School of Science, Changchun University of Science and Technology , Changchun, 130022 , P.R. China 1 School of Mathematics, Jilin University , Changchun, 130012 , P.R. China The aim of this paper is to deal with the existence and nonexistence of weak solutions to the initial and boundary value problem for ut = div(|?u|p(x,t)-2?u + b(x, t)?u) + f (u). By constructing suitable function spaces and applying the method of Galerkin's approximation as well as weak convergence techniques, the authors prove the existence of local solutions. Furthermore, we choose a suitable test-function, make integral estimates, and apply Gronwall's inequality to prove the uniqueness of weak solutions. At the end of this paper, the authors construct a suitable energy functional, obtain a new energy inequality, and apply a convex method to prove the nonexistence of solutions. Especially, it is worth pointing out that the results are obtained with the assumption that pt(x, t) is only negative and integrable, which is weaker than most of the other papers required. nonstandard growth condition; nonexistence of solutions; Galerkin's approximation - with logarithmic module of continuity: ? 2016 Gao et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. 1 Introduction Consider the following initial and boundary value problem: u(x, t) = ?, ??? u(x, ?) = u?(x), ?? ut = div(|?u|p(x,t)???u + b(x, t)?u) + f (u), (x, t) ? ?? (x, t) = ? x ? , ? (?, T ) := QT , ? (?, T ) := T , tinuous function satisfying ?z = (x, t), ? = (y, s) ? QT , |z ? ? | < ?, p(z) ? p(? ) ? ? |z ? ? | , li?m?s?u+p ?(? ) ln ?? = C < +?, and the coefficient b(x, t) is a Carath?odory function. Model (?.?) proposed by R??i?ka may describe some properties of electro-rheological fluids which change their mechanical properties dramatically when an external electric field is applied [?, ?]. The variable exponent p in Model (?.?) is a function of the external electric field |E|? which is subject to the quasi-static Maxwell equations Curl(E) = ?, where ?? is the dielectric constant in vacuum and the electric polarization P is linear in E, i.e. P = ?E. Another important application is the image processing where the anisotropy and nonlinearity of the diffusion operator are used to underline the borders of the distorted image and to eliminate the noise [?, ?]. For more physical background, the interested reader may refer to [????]. In the case when p(x, t) is a fixed constant, there have been many results about the existence, uniqueness, nonexistence, extinction of the solutions [??, ??, ??]. For nonconstant case, the authors of [???] and the authors of [??] studied the existence and uniqueness of weak solutions of the initial and Dirichlet boundary value problem with variable exponent of nonlinearity. Besides, the authors of [??] applied the differential and variational techniques to prove the existence of solutions when the exponent p only depends on the spatial variable. Motivated by the work above, we consider the existence and uniqueness of solutions to Problem (?.?). However, since the coefficient b(x, t) is degenerate or singular, it is natural to ask: which kind of conditions on the coefficient b(x, t) guarantees that Problem (?.?) admits a local solution? In our paper, we construct suitable function spaces and apply Galerkin?s method to prove the existence of weak solutions to Problem (?.?) with necessary uniform estimates and compactness argument. In addition, there exist some difficulties such as the failure of the monotonicity of the energy functional, the anisotropy of the diffusive operator and the gap between the norm and the modular, which make the methods used in [??] fail. In order to overcome such difficulties, we have to search a new technique or method. In this paper, by constructing a revised energy functional and combining a new energy estimate with convex method, we obtain the nonexistence of weak solutions when the exponents p is a function with respect to time and spatial variables. Especially, it is worth pointing out that the results are obtained with the assumption that pt(x, t) is only negative and integrable which is weaker than those the most of the other papers required. The outline of this paper is the following: In Section ?, we shall introduce the function spaces of Orlicz-Sobolev type, give the definition of the weak solution to the problem, and prove the existence of weak solutions with Galerkin?s method and the uniqueness of the solution. In Section ?, we establish sufficient condition of nonexistence of weak solutions 2 Existence of local solutions u Vt( ) = u ?, + ?u p(?), ; u H(QT ) = u ?,QT + ?u p(?),QT , |u|p(?) dx < ? , u p(?) = inf ? > ?, Ap(?)(u/?) ? ? ; u W?,p(?)( ) = u p(?), + ?u p(?), ; and denote by H (QT ) the dual of H(QT ) with respect to the inner product in L?(QT ). From [??], we know that condition (?.?) implies that M = {u : u ? W ?,p(x)( ), u = ? on ? } is equivalent to W??,p(x)( ) (the closure of C??( ) in W ?,p(x)( )). u p(?) < ? (= ?; > ?) u p(?) ? ? ? Ap(?)(u) < ? (= ?; > ?); + ? u pp(?) ? Ap(?)(u) ? u pp(?); ? + u pp(?) ? Ap(?)(u) ? u pp(?); u p(?) ? ? Ap(?)(u) ? ?; Ap(?)(u) ? ?. uv dx ? u p(?) v q(?) ? ? u p(?) v q(?). Because of the degeneracy, Problem (?.?) does not admit classical solutions in general. So we introduce weak solutions in the following sense. Definition ?.? A function u(x, t) ? H(QT ) ? L?(?, T ; L?( )), b(x, t)|?u| ? L?(?, T ; L?( )) is called a weak solution of Problem (?.?) if for every test-function |?u|p(x,t)???u?? dx dt < ?, t? t? On the other hand, b(x, t)|?u| ? L? ?, T ; L?( ) , which imply that the definition of weak solutions is well defined. The main theorem in this section is the following. Theorem ?.? Suppose that the continuous function f (s) and the exponents p(x, t), ? satisfy conditions (?.?)-(?.?). If the following conditions hold: Corollary ?.? Let the conditions of Theorem ?.? be fulfilled, then the solution u ? H(QT ) to Problem (?.?) satisfies the identity = ?, R (H?) the function f (s) is decreasing in s ? . Furthermore, we have the following comparison theorem. Corollary ?.? (Comparison principle) Let u, v ? H(QT ) ? L?(?, T ; H??( )) be two bounded weak solutions of Problem (?.?) such that u(x, ?) ? v(x, ?) a.e. in . If the nonlinearity exponents and the function f (s) satisfy the conditions of Theorem ?.?, then u(x, t) ? v(x, t) a.e. in QT . 3 Nonexistence of global weak solutions In this section, we concentrate on the study of nonexistence of weak solutions to Problem (?.?). For convenience, we first state that the function f (s) and the coefficient b(x, t) satisfy the following conditions: b(x, t) ? ?, f (u) ? C(R), bt(x, t) ? ?, ?(x, t) ? QT ; f (u)u ? p+G(u) ? ?, Definition ?.? A function u(x, t) is called a global solution to Problem (?.?) if ?T > ? the following property holds: Otherwise, we say that Problem (?.?) does not admit global weak solutions. Theorem ?.? Assume that u(x, t) ? H(QT ) ? L?(?, T ; L?( )), b(x, t)|?u| ? L?(?, T ; L?( )) is the local solution to Problem (?.?). If (?.?) and (?.?) are fulfilled and u? ? W??,p(x)( ), p+ > ?, such that G(u?) dx > |?u?|? dx + To prove Theorem ?.?, we need the following lemma. Lemma ?.? Assume that u ? H(QT ) is the solution to Problem (?.?), then u(x, t) satisfies the following relation: ?u(x, t) p(x) dx + ?u?(x) p(x) dx + G u(x, t) dx G u?(x) dx. Proof Following the lines of the proof of Lemma ?.? and Theorem ?.? in [?], we know that ut ? L?(QT ). Noting that = |?u|p(x)???u?ut, and using the idea of the proof of Lemma ? in [?], we arrive at the relation After integrating over (?, t), it is obvious that Lemma ?.? holds. Proof of Theorem ?.? Let t? = K (t) = ?b(x, ? )|?u|? ? |?u|p(x) + f (u)u dx + ??. By H?lder?s inequality, we have u?? dx = ? K (t) ? Therefore by Lemma ?.?, we obtain the following inequality: Thus by Schwarz?s inequality and the definition of K (t), we have bt|?u|? K (t)K (t) ? ? K (t) ? ? |?u|? dx + ? ? |?u|p(x) dx uf (u) ? p G(u) dx ? ?? p+ ? ? + ?G(u?) ? b(x, ?)|?u?|? ? |?u?| Noticing p+ > ?, K (t) > ?, we conclude from (?.?), (?.?), (?.?) that K (t)K (t) ? ? ?, for t ? (?, T ), which implies ? ?, for t ? (?, T ). ? T . This completes the proof of Theorem ?.?. On one hand, a simple analysis shows that pt = |?u|p(x,t)???u?ut + p? |?u|p(x,t) ln |?u|p(x,t) ? ? , ?u(x, t) p(x,t) dx + ?u?(x) p(x,?) dx + b(x?, t) ?u(x, t) ? dx ? G u(x, t) dx G u?(x) dx |?u|p(x,t) {|?u|p?e} p?(x, t) ln |?u|p(x,t) ? ? pt(x, t) dx dx ? The second inequality above follows from ? ?e ? s ln s ? ?, ? ? s ? ?. Lemma ?.? follows from (?.?) and (?.?). Our main result is as follows. G(u?) dx > |?u?|? dx + |?u?|p(x,?) dx then there exists T ? ? (?, T ] such that Proof We argue by contradiction. Define It is easy to verify that t? = K (t) = K (t) ? ? K (t) ? Therefore by Lemma ?.? and (?.??), we obtain the following inequality: ? K (t) ? ? |?u|p(x,t) dx ?G(u?) ? b(x, ?)|?u?|? ? |?u?|p(x,?) dx weaker than that of [?, ?]. putation shows that pt (x, y, z, t) = pt (x, y, z, t) dx dy dz dt = pt (x, y, z, t) ?/ L?. Competing interests The authors declare that they have no competing interests. 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Yanchao Gao, Ying Chu, Wenjie Gao. Existence,uniqueness, and nonexistence of solutions to nonlinear diffusion equations with p ( x , t ) -Laplacian operator, Boundary Value Problems, 2016, 149, DOI: 10.1186/s13661-016-0657-9