- Research
- Open access
- Published:
New method for the existence and uniqueness of solution of nonlinear parabolic equation
Boundary Value Problems volume 2015, Article number: 88 (2015)
Abstract
There are two contributions in this paper. The first is that the abstract result for the existence of the unique solution of certain nonlinear parabolic equation is obtained by using the properties of H-monotone operators, consequently, the proof is simplified compared to the corresponding discussions in the literature. The second is that the connections between resolvent of H-monotone operators and solutions of nonlinear parabolic equations are shown, and this strengthens the importance of H-monotone operators, which have already attracted the attention of mathematicians because of the connections with practical problems.
1 Introduction and preliminaries
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].
In 2008, Wei and Agarwal [18] studied the following nonlinear elliptic boundary value problem involving the generalized p-Laplacian:
where \(0\leq C(x)\in L^{p}(\Omega)\), ε is a non-negative constant and ϑ denotes the exterior normal derivative of Γ. It is shown that (1.1) has solutions in \(L^{s}(\Omega)\) under some conditions, where \(\frac{2N}{N+1} < p\leq s <+\infty\), \(1\leq q<+\infty\) if \(p\geq N\), and \(1\leq q \leq\frac{Np}{N-p}\) if \(p< N\), for \(N \geq1\). We observe that the proof, which uses Theorem 1.1 (stated in Section 1.2) as the main tool, is very complicated, since one needs to check that conditions (1.10) and (1.11) and the compactness of \(A+C_{1}\) are satisfied.
In 2010, Wei et al. [19] extended the work on elliptic equation to the following nonlinear parabolic equation involving the generalized p-Laplacian with mixed boundary conditions:
Some new technique has been used to tackle the existence of solutions of (1.2); specifically, the problem is divided into the following two auxiliary equations: (i) a parabolic equation with Dirichlet boundary conditions (1.3), and (ii) a parabolic equation with Neumann boundary value conditions (1.4):
By using Theorems 1.2 and 1.3 (stated in Section 1.2), it is shown that (1.3) has a unique solution. By employing Theorem 1.4, it is proved that (1.4) has a unique solution in \(L^{p}(0,T; W^{1,p}(\Omega))\), which implies that (1.2) has a unique solution in \(L^{p}(0,T; W^{1,p}(\Omega))\), where \(2 \leq p <+\infty\). However, we observe that the inequality (1.12) is not easy to check during the discussion.
Motivated by the work of Kawohl et al. [5, 9, 10, 13], Serrin et al. [3, 4, 12, 14] as well as Wei et al. [18, 19], in this paper we shall consider the following parabolic problem:
where \(\alpha: \mathbb {R}^{+} \cup\{0\} \rightarrow \mathbb {R}^{+}\) is a continuous nonlinear mapping such that \(pt \alpha'(t) + (p-1)\alpha(t)>0\), \(\alpha(t) \leq k_{1}\), for \(t \geq0\), \(\lim_{t \rightarrow +\infty}\alpha(t) = k_{2}> 0\), here \(k_{1}\) and \(k_{2}\) are positive constants.
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.
Example 1.1
If we set \(\alpha(t) = 1+ t (1+t^{2} )^{-\frac{1}{2}}\), \(t \geq0\), then it is obvious that \(\alpha: \mathbb {R}^{+}\cup\{0\} \rightarrow \mathbb {R}^{+}\) is a continuous nonlinear mapping, \(\alpha(t) \leq2 \) and \(\lim_{t \rightarrow +\infty}\alpha(t) = 2\). Moreover,
So, if \(\lambda_{1} \equiv\lambda_{2} \equiv\lambda\), then (1.5) becomes the following parabolic capillarity equation:
Example 1.2
For \(1 < p \leq2\), if we set \(\alpha(t) = (C+t^{\frac{2}{p}} )^{\frac{p-2}{2}}t^{\frac{2-p}{p}}\), \(t > 0\), where \(C \geq0\), then it is obvious that \(\alpha: \mathbb {R}^{+} \rightarrow \mathbb {R}^{+}\) is a continuous nonlinear mapping, \(\alpha(t) \leq1\) and \(\lim_{t \rightarrow+\infty}\alpha(t) = 1\). Moreover,
If \(\lambda_{2} \equiv0\), then (1.5) becomes the following parabolic equation with generalized p-Laplacian:
Example 1.3
If, in (1.7), \(C(x) \equiv0\), then (1.7) becomes the following parabolic p-Laplacian equation:
Example 1.4
For \(s \leq0\), if we set \(\alpha(t) = (1+t^{\frac{2}{p}} )^{\frac{s}{2}}t^{\frac{m-p+1}{p}}\), \(t > 0\), where \(m \geq0\), \(m+s+1 = p\), then it is obvious that \(\alpha: \mathbb {R}^{+} \rightarrow \mathbb {R}^{+}\) is a continuous nonlinear mapping, \(\alpha(t) \leq1 \) and \(\lim_{t \rightarrow+\infty}\alpha(t) = 1\). Moreover,
So, if \(\lambda_{2} \equiv0\), then (1.5) becomes the following parabolic curvature equation:
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].
Let \(J_{r}\) denote the duality mapping from X into \(2^{X^{*}}\), which is defined by
where \(r > 1\) is a constant. We use J to denote the usual normalized duality mapping. It is known that, in general, \(J_{r}(x) = \|x\|^{r-2}J(x)\), for all \(x \neq0\). Since \(X^{*}\) is strictly convex, J is a single-valued mapping [23].
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 \(\lim_{n\rightarrow +\infty}\|x_{n}\|= +\infty\).
Let \(B: X \rightarrow2^{X^{*}} \) be a maximal monotone operator such that \([0,0] \in G(B)\), then the equation \(J(u_{t} - u)+ t Bu_{t} \ni0\) has a unique solution \(u_{t} \in D(B)\) for every \(u \in X\) and \(t > 0\). The resolvent \(J_{t}^{B}\) and the Yosida approximation \(B_{t}\) of B are defined by [24]
for every \(u \in X\) and \(t > 0\). (Hence, \([J_{t}^{B} u, B_{t} u]\in G(B)\).)
Definition 1.1
([24])
Let C be a closed convex subset of X and let \(A: C \rightarrow 2^{X^{*}}\) be a multi-valued mapping. Then A is said to be a pseudo-monotone operator provided that
-
(i)
for each \(x \in C\), the image Ax is a non-empty closed and convex subset of \(X^{*}\);
-
(ii)
if \(\{x_{n}\}\) is a sequence in C converging weakly to \(x \in C \) and if \(f_{n} \in Ax_{n}\) is such that \(\limsup_{n\rightarrow \infty}(x_{n} - x,f_{n}) \leq0\), then to each element \(y \in C\), there corresponds an \(f(y) \in Ax \) with the property that \((x - y, f(y))\leq\liminf_{n \rightarrow\infty} (x_{n} - x, f_{n})\);
-
(iii)
for each finite-dimensional subspace K of X, the operator A is continuous from \(C \cap K\) to \(X^{*}\) in the weak topology.
Definition 1.2
Let \(\mathcal{H}\) be a Hilbert space. Let \(H: \mathcal{H} \rightarrow\mathcal{H}\) be a single-valued mapping and \(A : {\mathcal{H}} \rightarrow 2^{\mathcal{H}}\) be a multi-valued mapping. We say that A is H-monotone if A is monotone and \(R(H+\lambda A)(\mathcal{H}) = \mathcal{H}\), for every \(\lambda > 0\).
Lemma 1.1
([24])
If \(A: X \rightarrow2^{X^{*}}\) is a everywhere defined, monotone, and hemi-continuous mapping, then A is maximal monotone. If, moreover, A is coercive, then \(R(A)= X^{*}\).
Lemma 1.2
([24])
If \(\Phi: X \rightarrow (-\infty, +\infty]\) is a proper convex and lower-semicontinuous function, then ∂Φ is maximal monotone from X to \(X^{*}\).
Lemma 1.3
([24])
If \(A_{1}\) and \(A_{2}\) are two maximal monotone operators in X such that \((\operatorname{int}D(A_{1})) \cap D(A_{2})\neq \emptyset\), then \(A_{1} + A_{2}\) is maximal monotone.
Theorem 1.1
([27])
Let X be a real Banach space with a strictly convex dual space \(X^{*}\). Let \(J :X\rightarrow X^{*}\) be a duality mapping on X and there exists a function \(\eta : X \rightarrow[0,+\infty)\) such that for all \(u, v \in X\),
Let \(A, C_{1}:X\rightarrow2^{X}\) be accretive mappings such that
-
(i)
either both A and \(C_{1}\) satisfy condition (1.11), or \(D(A)\subset D(C_{1})\) and \(C_{1}\) satisfies condition (1.11):
$$\begin{aligned} \left \{ \begin{array}{@{}c@{}} \textit{for }u\in D(A)\textit{ and }v \in Au, \textit{there exists a constant }C(a,f)\textit{ such that}\\ \bigl(v - f, J(u - a)\bigr) \geq C(a,f). \end{array} \right . \end{aligned}$$(1.11) -
(ii)
\(A+C_{1}\) is m-accretive and boundedly-inversely-compact.
Let \(C_{2}:X\rightarrow X\) be a bounded continuous mapping such that, for any \(y\in X\), there is a constant \(C(y)\) satisfying \((C_{2}(u+y),Ju)\ge-C(y)\) for any \(u\in X\). Then the following results hold:
-
(a)
\(\overline{[R(A)+R(C_{1})]}\subset\overline {R(A+C_{1}+C_{2})}\);
-
(b)
\(\operatorname{int}[R(A)+R(C_{1})]\subset \operatorname{int} R(A+C_{1}+C_{2})\).
Theorem 1.2
([28])
Let \(T: X \rightarrow X^{*}\) be a bounded and pseudo-monotone operator, K be a closed and convex subset of X. Suppose that Φ is a lower-semicontinuous and convex function defined on K which is not always +∞ such that \(\Phi(v) \in(-\infty, +\infty]\) for all \(v \in K\). Suppose there exists \(v_{0} \in K\) such that \(\Phi(v_{0})<+\infty\) and
Then there exists \(u \in K\) such that
Theorem 1.3
([29])
Let X be a real reflexive Banach space with both X and its dual \(X^{*}\) being convex spaces. Let \(S :D(S) \subset X\rightarrow X^{*}\) be a linear maximal monotone operator and \(T: X \rightarrow X^{*}\) be a pseudo-monotone and coercive operator. Then, for each \(f \in X^{*}\), there exists an \(u \in D(S)\) such that, in the weak sense, \(S u + T u = f\).
Theorem 1.4
([30])
Let X be a real reflexive Banach space with both X and its dual \(X^{*}\) being strictly convex. Let J be the normalized duality mapping. Let A and B be two maximal monotone operators in X. Suppose there exist \(0 \leq k <1\) and \(C_{1}, C_{2}> 0 \) such that
for \(v \in D(A)\), \(a \in Av\) and \(t > 0\), where \(B_{t}\) is the Yosida approximation of B. Then \(R(A)+R(B)\simeq R(A+B)\).
Theorem 1.5
([25])
Let \(A : \mathcal{H} \rightarrow \mathcal{H}\) be a maximal monotone operator and \(H : \mathcal{H} \rightarrow\mathcal{H}\) be a bounded, coercive, hemi-contiunuous, and monotone mapping. Then A is H-monotone.
2 Main results
In this paper, unless otherwise stated, we shall assume that
In (1.5), Ω is a bounded conical domain of a Euclidean space \(\mathbb {R}^{N}\) with its boundary \(\Gamma\in C^{1}\) [6], T is a positive constant, \(\lambda_{1}\), \(\lambda_{2}\) and ε are non-negative constants, and ϑ denotes the exterior normal derivative of Γ. We shall assume that Green’s formula is available.
Suppose that \(g:\Omega\times \mathbb {R}^{N+2} \rightarrow \mathbb {R}\) is a given function satisfying the following conditions:
-
(a)
Carathéodory’s conditions
$$\begin{aligned}& x \rightarrow g(x,r)\mbox{ is measurable on } \Omega, \quad\mbox{for all } r \in \mathbb {R}^{N+2}; \\& r \rightarrow g(x,r)\mbox{ is continuous on } \mathbb {R}^{N+2}, \quad\mbox{for almost all }x\in\Omega. \end{aligned}$$ -
(b)
Growth condition
$$g(x,s_{1},\ldots,s_{N+2})\leq h(x)+ k_{3} |s_{1}|^{\min \{p/p', 1 \}}, $$where \((s_{1}, s_{2}, \ldots, s_{N+2})\in \mathbb {R}^{N+2} \), \(h(x)\in L^{2}(\Omega) \cap L^{p'}(\Omega)\) and \(k_{3} \) is a positive constant.
-
(c)
Monotone condition g is monotone with respect to \(r_{1}\), i.e.,
$$\bigl(g(x,s_{1},\ldots,s_{N+2})-g(x,t_{1}, \ldots,t_{N+2})\bigr) (s_{1} - t_{1}) \geq0 $$for all \(x \in\Omega\) and \((s_{1},\ldots,s_{N+2}), (t_{1},\ldots,t_{N+2})\in \mathbb {R}^{N+2}\).
-
(d)
Coercive condition
$$g(x,s_{1},\ldots,s_{N+2})s_{1} \geq k_{4} s_{1}^{2}, $$where \(k_{4}\) is a fixed positive constant.
Now, we present our discussion in the sequel.
Lemma 2.1
([18])
Let \(X_{0}\) denote the closed subspace of all constant functions in \(W^{1,p}(\Omega)\). Let X be the quotient space \(W^{1,p}(\Omega)/X_{0}\). For \(u\in W^{1,p}(\Omega)\), define the mapping \(P:W^{1,p}(\Omega) \rightarrow X_{0}\) by
Then there is a constant \(k_{5}>0\) such that for all \(u\in W^{1,p}(\Omega)\),
Lemma 2.2
Define the mapping \(B: L^{p}(0,T; W^{1,p}(\Omega))\rightarrow L^{p'}(0,T;(W^{1,p}(\Omega))^{*})\) by
for any \(u,w\in L^{p}(0,T; W^{1,p}(\Omega))\). Then B is strictly monotone, pseudo-monotone, and coercive.
(Here, \(\langle\cdot,\cdot\rangle\) and \(|\cdot|\) denote the Euclidean inner-product and Euclidean norm in \(\mathbb {R}^{N}\), respectively.)
Proof
Step 1. B is everywhere defined.
For \(u,w \in L^{p}(0,T; W^{1,p}(\Omega))\), we find
Since \(W^{1,p}(\Omega)\hookrightarrow L^{p}(\Omega)\hookrightarrow L^{r_{1}}(\Omega)\) and \(W^{1,p}(\Omega)\hookrightarrow L^{p}(\Omega)\hookrightarrow L^{r_{2}}(\Omega)\), for \(v \in W^{1,p}(\Omega)\), we have \(\|v\|_{L^{r_{1}}(\Omega)} \leq k_{6} \|v\|_{W^{1,p}(\Omega)}\), \(\|v\|_{L^{r_{2}}(\Omega)} \leq k_{7} \|v\|_{W^{1,p}(\Omega)}\), where \(k_{6}\) and \(k_{7}\) are positive constants. Hence,
which implies that B is everywhere defined.
Step 2. B is strictly monotone.
For \(u, v \in L^{p}(0,T;W^{1,p}(\Omega))\), we have
If we set \(f(s) = s^{1-\frac{1}{p}}\alpha(s)\), \(s > 0\), then in view of the assumption of α, we have
which implies that f is strictly monotone. Hence, B is strictly monotone.
Step 3. B is hemi-continuous.
It suffices to show that for any \(u,v,w\in L^{p}(0,T;W^{1,p}(\Omega))\) and \(t\in[0,1]\), \((w,B(u+tv)-Bu) \rightarrow0\) as \(t\rightarrow 0\). Since
by Lebesque’s dominated convergence theorem and noting that α is continuous, we find
Hence, B is hemi-continuous.
Step 4. B is coercive.
We shall first show that for \(u\in L^{p}(0,T; W^{1,p}(\Omega))\),
where \(k_{8}\) and \(k_{9}\) are positive constants.
In fact, using Lemma 2.1, we know that, for \(u \in L^{p}(0,T; W^{1,p}(\Omega))\),
Thus,
Since
we have
Therefore,
If we set \(k_{8} = (k_{5}^{p} + 1 )^{\frac{1}{p}}\), then (2.1) is true.
Since \(\lim_{t \rightarrow+\infty}\alpha(t) = k_{2} > 0\), there exists sufficiently large \(K > 0\) such that \(\alpha(t)> \frac{l}{2}\) whenever \(t > K\). Now, for \(u\in L^{p}(0,T; W^{1,p}(\Omega))\), let \(\|u\|_{L^{p}(0,T; W^{1,p}(\Omega))}\rightarrow+\infty\). Using (2.1), we find
This completes the proof. □
Lemma 2.3
The mapping \(\Phi: L^{p}(0,T; W^{1,p}(\Omega))\rightarrow \mathbb {R}\) defined by
for any \(u \in L^{p}(0,T; W^{1,p}(\Omega))\), is proper, convex, and lower-semicontinuous on \(L^{p}(0,T; W^{1,p}(\Omega))\). Moreover, the subdifferential ∂Φ of Φ is maximal monotone in view of Lemma 1.2.
Proof
The proof is similar to that of Lemma 3.1 in [27]. □
Lemma 2.4
([19])
Define \(S: D(S) \rightarrow L^{p'}(0,T; (W^{1,p}(\Omega))^{*})\) by
where
The mapping S is linear maximal monotone.
Definition 2.1
Define a mapping \(A :L^{2}(0,T; L^{2}( \Omega))\rightarrow2^{L^{2}(0,T; L^{2}( \Omega))}\) by
for \(u\in D(A)= \{u\in L^{2}(0,T; L^{2}( \Omega)) \vert \mbox{ there exists a }w(x)\in L^{2}(0,T; L^{2}( \Omega))\mbox{ such that } w(x) \in Bu+\partial\Phi(u)+Su\}\).
Lemma 2.5
Define the mapping \(F : L^{p}(0,T; W^{1,p}(\Omega)) \rightarrow L^{p'}(0,T;(W^{1,p}(\Omega))^{*})\) by
for \(u(x,t), v(x,t) \in L^{p}(0,T; W^{1,p}(\Omega))\). Then F is everywhere defined.
Proof
Step 1. For \(u(x,t) \in L^{p}(0,T; W^{1,p}(\Omega))\), \(x \rightarrow g (x, u, \frac{\partial u}{\partial t}, \varepsilon\nabla u )\) is measurable on Ω.
From the fact that \(u(x,t), \frac{\partial u}{\partial x_{i}} \in L^{p}(\Omega)\), \(i = 1 ,2, \ldots, N\), we see that \(x \rightarrow (u, \frac{\partial u}{\partial x_{1}}, \ldots, \frac{\partial u}{\partial x_{N}} )\) is measurable on Ω. Combining with the fact that g satisfies Carathéodory’s conditions, we know that \(x \rightarrow g (x, u, \frac{\partial u}{\partial t}, \varepsilon \nabla u )\) is measurable on Ω.
Step 2. F is everywhere defined.
For \(u,v\in L^{p}(0,T; W^{1,p}(\Omega))\), we have
which implies that F is everywhere defined.
This completes the proof. □
Definition 2.2
Define the mapping \(H : L^{2}(0,T; L^{2}( \Omega))\rightarrow L^{2}(0,T; L^{2}( \Omega))\) by
for \(u \in D(H) = \{u(x) \in L^{2}(0,T; L^{2}( \Omega)) | \mbox{ there exists }v(x)\in L^{2}(0,T; L^{2}( \Omega)) \mbox{ such that }v(x) = Fu(x)\}\), where F is the same as in Lemma 2.5.
Lemma 2.6
The mapping \(H : L^{2}(0,T; L^{2}( \Omega))\rightarrow L^{2}(0,T; L^{2}( \Omega))\) defined in Definition 2.2 is bounded, coercive, hemi-continuous, and monotone.
Proof
Step 1. H is bounded.
From condition (b) of g, we know that
where \(k_{10}\) and \(k_{11}\) are positive constants. This implies that H is bounded.
Step 2. H is coercive.
From condition (d) of g, we know that
as \(\|u\|_{L^{2}(0,T; L^{2}( \Omega))} \rightarrow +\infty\). Hence, H is coercive.
Step 3. H is hemi-continuous.
Since g satisfies condition (a), we have, for any \(w(x,t) \in L^{2}(0,T; L^{2}( \Omega))\),
as \(t \rightarrow0\), which implies that H is hemi-continuous.
Step 4. H is monotone.
In view of condition (c) of g, we have
which implies that H is monotone.
This completes the proof. □
Lemma 2.7
For all \(u, v \in L^{p}(0,T; W^{1,p}(\Omega))\), we have
Moreover, \(0 \in\partial\Phi(0)\).
Proof
The idea of the proof mainly comes from Proposition 3.2(ii) in [27]. For completeness, we give the outline of the proof as follows.
Define the mapping \(G:L^{p}(0,T;L^{p}(\Gamma))\rightarrow L^{p'}(0,T; L^{p'}(\Gamma))\) by \(Gu = \beta_{x}(u) \), for any \(u\in L^{p}(0,T; L^{p}(\Gamma))\). Also, define the mapping \(K:L^{p}(0,T; W^{1,p}(\Omega))\rightarrow L^{p}(0,T; L^{p}(\Gamma)) \) by \(K(v)= v|_{\Gamma}\), for any \(v\in L^{p}(0,T; W^{1,p}(\Omega))\). Then \(K^{*}GK =\partial\Phi\), where Φ is the same as in Lemma 2.3.
In fact, it is obvious that G is continuous. For \(u(x,t), v(x,t) \in L^{p}(0,T; L^{p}(\Gamma))\), we have \((u - v, Gu - Gv)= \int_{0}^{T}\int_{\Gamma}(\beta_{x}(u)- \beta_{x}(v)) (u- v)\,d\Gamma(x)\,dt \geq 0\), since \(\beta_{x}\) is monotone. Thus, G is monotone. In view of Lemma 1.1, \(G : L^{p}(0,T; L^{p}(\Gamma))\rightarrow L^{p'}(0,T; L^{p'}(\Gamma))\) is maximal monotone.
Define \(\Psi: L^{p}(0,T; L^{p}(\Gamma)) \rightarrow \mathbb {R}\) by \(\Psi(u)= \int_{0}^{T} \int_{\Gamma}\varphi_{x}(u)\,d\Gamma(x)\,dt\). It is easy to see that Ψ is a proper, convex, and lower-semicontinuous function on \(L^{p}(0,T; L^{p}(\Gamma))\), which implies that \(\partial\Psi: L^{p}(0,T; L^{p}(\Gamma))\rightarrow L^{p'}(0,T; L^{p'}(\Gamma))\) is maximal monotone in view of Lemma 1.2. Since
for all \(u(x,t), v(x,t) \in L^{p}(0,T; L^{p}(\Gamma))\), we have \(Gv \in\partial\Psi(v)\). So \(G = \partial\Psi\).
Now, it is clear that \(K^{*}GK: L^{p}(0,T; W^{1,p}(\Omega))\rightarrow L^{p'}(0,T; (W^{1,p}(\Omega))^{*}) \) is maximal monotone since both K and G are continuous. Finally, for any \(u,v\in L^{p}(0,T; W^{1,p}(\Omega))\), we have
Hence, we get \(K^{*}GK \subset\partial\Phi\) and so \(K^{*}GK = \partial\Phi\).
It now follows that for all \(u, v \in L^{p}(0,T; W^{1,p}(\Omega))\),
Moreover, \(0 \in\partial\Phi(0)\) since \(0 \in\beta_{x}(0)\). This completes the proof. □
Lemma 2.8
The mapping \(A : L^{2}(0,T; L^{2}(\Omega)) \rightarrow L^{2}(0,T; L^{2}(\Omega))\) defined in Definition 2.1 is maximal monotone.
Proof
Noting Lemmas 2.2-2.4, we can easily get the result that A is monotone.
Next, we shall show that \(R(I+A) = L^{2}(0,T; L^{2}(\Omega))\), which ensures that A is maximal monotone.
Case 1. \(p \geq2\). We define \(\overline{F}:L^{p}(0,T; W^{1,p}(\Omega)) \rightarrow L^{p'}(0,T; (W^{1,p}(\Omega))^{*})\) by
where \((\cdot,\cdot)_{L^{2}(0,T; L^{2}(\Omega))}\) denotes the inner-product of \(L^{2}(0,T; L^{2}(\Omega))\). Then \(\overline{F}\) is everywhere defined, monotone and hemi-continuous, which implies that \(\overline{F}\) is maximal monotone in view of Lemma 1.1. Combining with the facts of Lemmas 1.3, 2.2-2.4, we have \(R(B+\partial\Phi+ S + \overline{F}) = L^{p'}(0,T; (W^{1,p}(\Omega))^{*})\).
For \(f \in L^{2}(0,T; L^{2}(\Omega)) \subset L^{p'}(0,T; (W^{1,p}(\Omega))^{*})\), there exists \(u \in L^{p}(0,T; W^{1,p}(\Omega ))\subset L^{2}(0,T; L^{2}(\Omega))\) such that
which implies that \(R(I+A)= L^{2}(0,T; L^{2}(\Omega))\).
Case 2. \(\frac{2N}{N+1} < p<1\), then \(p' \geq 2\). Similar to Lemma 2.2, we define \(\widehat{B}: L^{p'}(0,T; W^{1,p}(\Omega))\rightarrow L^{p}(0,T;(W^{1,p}(\Omega))^{*})\) by
for any \(u,w\in L^{p'}(0,T; W^{1,p}(\Omega))\). Then \(\widehat{B}\) is maximal monotone and coercive. Similar to Lemma 2.3, define the mapping \(\widehat{\Phi} : L^{p'}(0,T; W^{1,p}(\Omega))\rightarrow \mathbb {R}\) by
for any \(u \in L^{p'}(0,T; W^{1,p}(\Omega))\), then \(\partial\widehat{\Phi}\) is maximal monotone. Similar to Lemma 2.4, define \(\widehat{S}: D(\widehat{S})= \{ u\in L^{p'}(0,T; W^{1,p}(\Omega)) | \frac{\partial u}{\partial t}\in L^{p}(0,T; (W^{1,p}(\Omega))^{*}), u(x,0) = u(x,T) \} \rightarrow L^{p}(0,T; (W^{1,p}(\Omega ))^{*})\) by
Then \(\widehat{S}\) is linear maximal monotone. Similar to Case 1, define \(\overline{\overline{F}}:L^{p'}(0,T; W^{1,p}(\Omega)) \rightarrow L^{p}(0,T; (W^{1,p}(\Omega))^{*})\) by
then we have \(R(\widehat{B}+\partial\widehat{\Phi}+\widehat{S}+\overline {\overline{F}}) = L^{p}(0,T; (W^{1,p}(\Omega))^{*})\). So, for \(f \in L^{2}(0,T; L^{2}(\Omega)) \subset L^{p}(0,T; (W^{1,p}(\Omega))^{*})\), there exists \(u \in L^{p'}(0,T; W^{1,p}(\Omega ))\subset L^{2}(0,T; L^{2}(\Omega))\) such that
which implies that \(R(I+A)= L^{2}(0,T; L^{2}(\Omega))\). □
Theorem 2.1
For \(f(x,t) \in L^{2}(0,T; L^{2}(\Omega))\), the nonlinear parabolic equation (1.5) has a unique solution \(u(x,t) \) in \(L^{2}(0,T; L^{2}(\Omega))\), i.e.,
-
(a)
\(\frac{\partial u}{\partial t}-\operatorname{div} [\alpha(|\nabla u|^{p})|\nabla u|^{p-2}\nabla u ]+\lambda_{1} |u|^{r_{1}-2}u + \lambda_{2} |u|^{r_{2}-2}u + g (x,u,\frac{\partial u}{\partial t}, \varepsilon \nabla u ) = f(x,t)\), a.e. \((x,t)\in\Omega\times(0,T)\);
-
(b)
\(- \langle\vartheta,\alpha(|\nabla u|^{p})|\nabla u|^{p-2}\nabla u \rangle\in\beta_{x}(u(x,t))\), a.e. \(x\in \Gamma \times(0,T)\);
-
(c)
\(u(x,0) = u(x,T)\), \(x \in\Omega\).
Proof
We split our proof into two steps.
Step 1. There exists a unique \(u(x,t)\) which satisfies \(Hu+\lambda Au = f\), where \(f(x,t) \in L^{2}(0,T; L^{2}(\Omega))\) is a given function.
From Theorem 1.5, Lemmas 2.6 and 2.8, we know that A is H-monotone. Thus, \(R(H+\lambda A)= L^{2}(0,T; L^{2}(\Omega))\). Then, for \(f(x,t) \in L^{2}(0,T; L^{2}(\Omega))\) in (1.5), there exists \(u(x,t) \in L^{2}(0,T; L^{2}(\Omega))\) such that \(Hu(x,t)+\lambda Au(x,t) = f(x,t)\). Next, we shall prove that \(u(x,t)\) is unique.
Suppose that \(u(x,t)\) and \(v(x,t)\) satisfy \(Hu+\lambda Au = f\) and \(Hv + \lambda Av = f\), respectively. Then \(0 \leq\lambda(u - v, Au - Av) = - (u - v, Hu - Hv)\leq0\), which ensures that
Using Lemmas 2.2, 2.3, and 2.4, we have \((u - v, Bu - Bv)= 0\), which implies that \(u(x,t) = v(x,t)\), since B is strictly monotone.
Step 2. If \(u(x,t)\in L^{2}(0,T; L^{2}(\Omega))\) satisfies \(f = H u + Au\), then \(u(x,t)\) is the solution of (1.5).
Since \(\Phi(u+ \varphi) = \Phi(u)\) for any \(\varphi\in C^{\infty}_{0}(\Omega\times(0,T))\), we have \((\varphi, \partial\Phi(u)) = 0\). Then, for \(\varphi\in C_{0}^{\infty}(\Omega\times(0,T))\), we have
So
which implies that the equation
is true.
By using (2.2) and Green’s formula, we have
Then
From the definition of S, we can easily obtain \(u(x,0) = u(x,T)\) for all \(x \in\Omega\). Combining with (2.2) and (2.4) we see that u is the unique solution of (1.5).
This completes the proof. □
Lemma 2.9
Define \(\widetilde{B} : L^{p}(0,T; W^{1,p}(\Omega)) \rightarrow L^{p'}(0,T; (W^{1,p}(\Omega))^{*})\) by \(\widetilde{B}u \equiv Bu - f(x,t)\), for \(u \in L^{p}(0,T; W^{1,p}(\Omega))\). Then \(\widetilde{B}\) is maximal monotone.
Proof
Similar to the proof of Lemma 2.2, we know that \(\widetilde{B}\) is everywhere defined, monotone, and hemi-continuous. It follows that \(\widetilde{B}\) is maximal monotone. □
Definition 2.3
Define a mapping \(\widetilde{A} :L^{2}(0,T; L^{2}( \Omega))\rightarrow2^{L^{2}(0,T; L^{2}( \Omega))}\) by
for \(u\in D(\widetilde{A})= \{u\in L^{2}(0,T; L^{2}( \Omega)) | \mbox{there exists }w(x) \in L^{2}(0,T; L^{2}( \Omega ))\mbox{ such that }w(x) \in\widetilde{B}u +\partial\Phi(u)+Su\}\).
Definition 2.4
Let \(\mathcal{H}\) be a Hilbert space and A be a H-monotone operator. The resolvent operator of A, \(R_{A, \lambda}^{H} : \mathcal{H} \rightarrow\mathcal{H}\), is defined by
Theorem 2.2
\(u(x,t) = R^{H}_{\widetilde{A},1}(0)\) if and only if \(u(x,t) \in L^{2}(0,T; L^{2}( \Omega))\) is the solution of (1.5).
Proof
Let \(u(x,t)\) be the solution of (1.5). Then, using Green’s formula and Lemma 2.7, we have
Thus, \(u(x,t) = R^{H}_{\widetilde{A},1}(0)\).
If \(u(x,t) = R^{H}_{\widetilde{A},1}(0)\), then noting Lemma 2.7, we have for \(\varphi\in C^{\infty}_{0}(\Omega\times(0,T))\),
which implies that the equation
is true.
Similar to the last part of Theorem 2.1, we know that \(- \langle\vartheta, \alpha(|\nabla u|^{p})|\nabla u|^{p-2}\nabla u \rangle\in\beta_{x}(u(x,t))\). From the definition of S, we know that \(u(x,0) = u(x,T)\) for all \(x\in\Omega\), which implies that \(u(x,t)\) is the solution of (1.5). This completes the proof. □
References
Lieberman, GM: Second Order Parabolic Differential Equations. World Scientific, River Edge (1996)
Lieberman, GM: Oblique Derivative Problems for Elliptic Equations. World Scientific, Hackensack (2013)
Levine, HA, Park, SR, Serrin, J: Global existence and nonexistence theorems for quasilinear evolution equations of formally parabolic type. J. Differ. Equ. 142, 212-229 (1998)
García-Huidobro, M, Manásevich, R, Serrin, J, Tang, M, Yarur, CS: Ground states and free boundary value problems for the n-Laplacian in n dimensional space. J. Funct. Anal. 172, 177-201 (2000)
Fira, M, Kawohl, B: Large time behavior of solutions to a quasilinear parabolic equation with a nonlinear boundary condition. Adv. Math. Sci. Appl. 11, 113-126 (2001)
Wei, L, He, Z: The applications of theories of accretive operators to nonlinear elliptic boundary value problems in L p-spaces. Nonlinear Anal. 46, 199-211 (2001)
Wei, L, Zhou, HY: Existence of solutions of a family of nonlinear boundary value problems in \(L^{2}\)-spaces. Appl. Math. J. Chin. Univ. Ser. B 20, 175-182 (2005)
Wei, L, Zhou, HY: The existence of solutions of nonlinear boundary value problem involving the p-Laplacian operator in \(L^{s}\)-spaces. J. Syst. Sci. Complex. 18, 511-521 (2005)
Belloni, M, Kawohl, B, Juutinen, P: The p-Laplace eigenvalue problem as \(p\to\infty\) in a Finsler metric. J. Eur. Math. Soc. 8, 123-138 (2006)
Kawohl, B, Lindqvist, P: Positive eigenfunctions for the p-Laplace operator revisited. Analysis 26, 545-550 (2006)
Wei, L, Zhou, HY: Research on the existence of solution of equation involving p-Laplacian operator. Appl. Math. J. Chin. Univ. Ser. B 21, 191-202 (2006)
Farina, A, Serrin, J: Entire solutions of completely coercive quasilinear elliptic equations. J. Differ. Equ. 250, 4367-4408 (2011)
Kawohl, B: Variations on the p-Laplacian. In: Nonlinear Elliptic Partial Differential Equations. Contemp. Math., vol. 540, pp. 35-46. Am. Math. Soc., Providence (2011)
D’Ambrosio, L, Farina, A, Mitidieri, E, Serrin, J: Comparison principles, uniqueness and symmetry results of solutions of quasilinear elliptic equations and inequalities. Nonlinear Anal. 90, 135-158 (2013)
Amann, H: Periodic solutions of semilinear parabolic equations. In: Nonlinear Analysis (Collection of papers in honor of Erich H. Rothe), pp. 1-29. Academic Press, New York (1978)
Lieberman, GM: Time-periodic solutions of quasilinear parabolic differential equations. III. Conormal boundary conditions. Nonlinear Anal. 45, 755-773 (2001)
Fife, P: Solutions of parabolic boundary problems existing for all time. Arch. Ration. Mech. Anal. 16, 155-186 (1964)
Wei, L, Agarwal, RP: Existence of solutions to nonlinear Neumann boundary value problems with generalized p-Laplacian operator. Comput. Math. Appl. 56, 530-541 (2008)
Wei, L, Agarwal, RP, Wong, PJY: Existence of solutions to nonlinear parabolic boundary value problems with generalized p-Laplacian operator. Adv. Math. Sci. Appl. 20, 423-445 (2010)
Wei, L, Duan, LL, Zhou, HY: Study on the existence and uniqueness of solution of generalized capillarity problem. Abstr. Appl. Anal. 2012, Article ID 154307 (2012)
Wei, L, Tan, RL, Fan, SX: Existence of solutions of one kind curvature equation with two kinds of boundary value conditions. Math. Appl. 27, 131-139 (2014)
Barbu, V: Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoff, Leyden (1976)
Takahashi, W: Nonlinear Functional Analysis - Fixed Point Theory and Its Applications. Yokohama Publishers, Yokohama (2000)
Pascali, D, Sburlan, S: Nonlinear Mappings of Monotone Type. Sijthoff & Noordhoff, Rockville (1978)
Fang, YP, Huang, NJ: H-Accretive operators and resolvent operator technique for solving variational inclusions in Banach spaces. Appl. Math. Lett. 17, 647-653 (2004)
Ding, XP: Parametric completely generalized mixed implicit quasi-variational inclusions involving h-maximal monotone mappings. J. Comput. Appl. Math. 18, 252-269 (2005)
Calvert, BD, Gupta, CP: Nonlinear elliptic boundary value problems in L p-spaces and sums of ranges of accretive operators. Nonlinear Anal. 2, 1-26 (1978)
Brezis, H: Équations et inéquations non linéaires dans les espaces vectoriels en dualité. Ann. Inst. Fourier (Grenoble) 18, 115-175 (1968)
Chen, ZC, Luo, T: The initial-boundary value problem for quasilinear integro-differential equations. Int. J. Differ. Equ. Appl. 6, 299-306 (2002)
Reich, S: The range of sums of accretive and monotone operators. J. Math. Anal. Appl. 68, 310-317 (1979)
Acknowledgements
Li Wei is supported by the National Natural Science Foundation of China (11071053), Natural Science Foundation of Hebei Province (No. A2014207010), Key Project of Science and Research of Hebei Educational Department (ZH2012080) and Key Project of Science and Research of Hebei University of Economics and Business (2013KYZ01).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Rights and permissions
Open Access 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.
About this article
Cite this article
Wei, L., Agarwal, R.P. & Wong, P.J. New method for the existence and uniqueness of solution of nonlinear parabolic equation. Bound Value Probl 2015, 88 (2015). https://doi.org/10.1186/s13661-015-0343-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13661-015-0343-3