New method for the existence and uniqueness of solution of nonlinear parabolic equation

1.1 Introduction

Nonlinear boundary value problems involving the generalized p-Laplacian operator arise from many physical phenomena, such as reaction-diffusion problems, petroleum extraction, flow through porous media and non-Newtonian fluids, just to name a few. Thus, the study of such problems and their generalizations have attracted numerous attention in recent years. In particular, we would mention the books of Lieberman [1, 2] where in [1] the theory of linear and quasilinear parabolic second-order partial differential equations is elaborated, with emphasis on the Cauchy-Dirichlet problem and the oblique derivative problem in bounded space-time domains; while in [2] a detailed qualitative analysis of second-order elliptic boundary value problems that involve oblique derivatives is presented. A sample of other research work that contributes to the literature of parabolic and elliptic problems includes [3–14] listed chronologically as well as the references cited therein. For time-periodic case which is the concern of this paper, we refer the reader to [15–17].

Let \(\varphi:\Gamma\times \mathbb {R}\rightarrow \mathbb {R}\) be a given function such that, for each \(x\in\Gamma\), \(\varphi_{x}= \varphi(x,\cdot):\mathbb {R}\rightarrow \mathbb {R}\) is a proper, convex and lower-semicontinuous function with \(\varphi_{x}(0)=0\). Let \(\beta_{x}\) be the subdifferential of \(\varphi_{x}\), i.e., \(\beta_{x}\equiv\partial\varphi_{x}\). Suppose that \(0\in\beta_{x}(0)\) and for each \(t\in \mathbb {R}\), the function \(x\in\Gamma\rightarrow (I+\lambda\beta_{x})^{-1}(t)\in \mathbb {R}\) is measurable for \(\lambda>0\). More details of (1.5) will be presented in Section 2.

There are some major differences between parabolic problems (1.2) and (1.5): (i) The main part \(- \operatorname{div} [\alpha(|\nabla u|^{p})|\nabla u|^{p-2}\nabla u ]\) in (1.5) includes the main part \(-\operatorname{div} [(C(x,t)+|\nabla u|^{2})^{\frac{p-2}{2}}\nabla u ]\) in (1.2); (ii) the term \(g (x,u,\frac{\partial u}{\partial t}, \varepsilon \nabla u )\) is considered in (1.5) but not in (1.2); (iii) \(\beta_{x}(u(x,t))\) in (1.5) is different from \(\beta (u)-h(x,t)\) in (1.2).

The existence of the unique solution of (1.5) will be discussed in \(L^{2}(0,T; L^{2}(\Omega))\), which does not change while p is varying from \(\frac{2N}{N+1}\) to +∞ for \(N \geq1\). Hence, the result is different from that on (1.2) in [19]. Our main tool in this paper will be Theorem 1.5 (stated in Section 1.2). Consequently, the proof of our result is different from and comparatively simplified with respect to that of [19].

Actually, (1.5) is very general and it includes the following special cases. The related work can be found in [19–21] and the references cited therein.

1.2 Preliminaries

Let X be a real Banach space with a strictly convex dual space \(X^{*}\). We shall use \((\cdot,\cdot)\) to denote the generalized duality pairing between X and \(X^{*}\). We shall use “→” and “\(w\mbox{-}\!\lim\)” to denote strong and weak convergence, respectively. Let “\(X \hookrightarrow Y\)” denote the space X embedded continuously in space Y. For any subset G of X, we denote by intG its interior and \(\overline{G}\) its closure, respectively. For two subsets \(G_{1}\) and \(G_{2}\) in X, if \(\overline{G_{1}}= \overline{G_{2}}\) and \(\operatorname{int} G_{1} = \operatorname{int} G_{2}\), then we say \(G_{1}\) is almost equal to \(G_{2}\), which is denoted by \(G_{1} \simeq G_{2}\). A mapping \(T:X \rightarrow X^{*}\) is said to be hemi-continuous on X [22] if \(w\mbox{-}\!\lim_{t \rightarrow0}T(x+ty) = Tx\) for any \(x,y \in X\).

A function Φ is called a proper convex function on X [22] if Φ is defined from X to \((-\infty, +\infty]\), not identically +∞, such that \(\Phi((1-\lambda)x+\lambda y)\leq(1-\lambda)\Phi(x)+\lambda\Phi(y)\), whenever \(x , y \in X\) and \(0 \leq\lambda\leq1\).

A function \(\Phi: X \rightarrow(-\infty, +\infty]\) is said to be lower-semicontinuous on X [22] if \(\liminf_{y \rightarrow x}\Phi(y)\geq\Phi(x)\), for any \(x \in X\).

Given a proper convex function Φ on X and a point \(x \in X\), we denote by \(\partial\Phi(x)\) the set of all \(x^{*} \in X^{*}\) such that \(\Phi(x)\leq\Phi(y)+ (x - y, x^{*})\), for every \(y \in X\). Such element \(x^{*}\) is called the subgradient of Φ at x, and \(\partial\Phi(x)\) is called the subdifferential of Φ at x [22].

A multi-valued mapping \(A:X\rightarrow2^{X}\) is said to be accretive [22, 23] if \((v_{1}-v_{2},J_{r}(u_{1}-u_{2}))\ge0\), for any \(u_{i} \in D(A)\) and \(v_{i}\in Au_{i}\), \(i=1,2\). The accretive mapping A is said to be m-accretive if \(R(I+\lambda A)=X\) for some \(\lambda>0\).

A multi-valued operator \(B:X\rightarrow2^{X^{*}}\) is said to be monotone [24] if its graph \(G(B)\) is a monotone subset of \(X\times X^{*}\) in the sense that \((u_{1}-u_{2},w_{1}-w_{2})\ge 0\), for any \([u_{i},w_{i}]\in G(B)\), \(i=1,2\). Further, B is called strictly monotone if \((u_{1}-u_{2},w_{1}-w_{2})\ge0\) and the equality holds if and only if \(u_{1} = u_{2}\). The monotone operator B is said to be maximal monotone if \(G(B)\) is maximal among all monotone subsets of \(X\times X^{*}\) in the sense of inclusion. Also, B is maximal monotone if and only if \(R(B + \lambda J) = X^{*}\), for any \(\lambda > 0\). The mapping B is said to be coercive [24] if \(\lim_{n\rightarrow+\infty}{(x_{n},x^{*}_{n})}/{\|x_{n}\|}=+\infty\) for all \([x_{n},x^{*}_{n}]\in G(B)\) such that \(\lim_{n\rightarrow +\infty}\|x_{n}\|= +\infty\).