Elliptic differential equations, parabolic differential equations and hyperbolic differential equations are three kinds of important differential equations. Inspired by Calvert and Gupta's perturbation result on the ranges of nonlinear $ m $ -accretive mappings presented in [1], the elliptic $ p $ -Laplacian boundary value problems and their general forms were extensively studied in work of [2-6]. Actually, [6] can be regarded as the summary of the work done in [2-5]. Namely, the following elliptic equation involving the generalized $ p $ -Laplacian operator with Neumann boundaries is studied

$\left\{ \begin{aligned} & -{\rm div}[(C(x)+|\nabla u|^2) ^{\frac{p-2}{2}}\nabla u]+\varepsilon |u|^{q-2}u+g(x,u(x))= f(x) \mbox{ a.e. in}\ \Omega,\\ & -\langle \vartheta,(C(x)+|\nabla u|^2) ^{\frac{p-2}{2}}\nabla u \rangle \in \beta_{x}(u(x)) \mbox{a.e. on}\ \Gamma,\end{aligned} \right.$

(1.1)

where $ \beta_x $ is the subdifferential of a proper, convex and lower-semi continuous function. It was shown in [6] 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 \geq 1. $

Recently, the work done in [2-6] is extended to the following one

$\left\{ \begin{aligned} & -{\rm div}(\alpha(|\nabla u|^p)|\nabla u|^{p-2}\nabla u)+λ_1|u|^{q-2}u+λ_2|u|^{r-2}u+g(x,u(x),\varepsilon\nabla u(x))= f(x),\\ & -<\vartheta,\alpha(|\nabla u|^p)|\nabla u|^{p-2}\nabla u> \in \beta_{x}(u(x)) \mbox{a.e. on}\ \Gamma. \end{aligned} \right.$

(1.2)

By using the properties of $ H $ -accretive mappings, it is shown in[7] that (1.2) has solutions in $ L^2(\Omega) $ under some conditions, where $ \frac{2N}{N+1} < p <+\infty,1\leq q,r<+\infty $ if $ p\geq N,$ and $ 1\leq q,r \leq \frac{Np}{N-p} $ if $ p<N $ for $ N \geq 1. $

As for parabolic differential equation, Wei and Agarwal [8]studied the following one

$\left\{ \begin{aligned} & \frac{\partial u}{\partial t}-{\rm div}(\alpha(\nabla u)) + \varepsilon |u|^{q-2}u = f(x,t),(x,t) \in \Omega × (0,T),\\& -\langle\vartheta,\alpha(\nabla u)\rangle \in \beta(u(x,t))-h(x,t),(x,t) \in \Gamma × (0,T),\\& u(x,0) = u(x,T),x \in \Omega. \end{aligned} \right.$

(1.3)

By using some results on the ranges for bounded pseudo-monotone operator and maximal monotone operator presented in [9, 10], they obtained that (1.3) has solutions in $ L^p(0,T; W^{1,p}(\Omega)) $ for $ 1 < q \leq p <+\infty. $

How about hyperbolic differential equations? Can we use the perturbation theories for nonlinear operators to do the analysis?

In this paper, we shall study the following hyperbolic problem with mixed boundaries

$\left\{ \begin{aligned} & -\frac{\partial }{\partial t}(\alpha_1(\frac{\partial u}{\partial t })) - {\rm div}(\alpha_2(|\nabla u|^p)|\nabla u|^{p-2}\nabla u) + λ_1|u|^{r_1-2}u+λ_2|u|^{r_2-2}u = f(x,t),\\ &(x,t) \in \Omega × (0,T),\\ & -\langle\vartheta,\alpha_2(|\nabla u|^p)|\nabla u|^{p-2}\nabla u\rangle \in \beta(u(x,t))-h(x,t),(x,t) \in \Gamma × (0,T),\\ & \gamma u(x,t) = w(x,t) {\rm a.e.} (x,t) \in \Gamma × (0,T),\\& u(x,0) = u(x,T),x \in \Omega,\\ & \alpha_1(\frac{\partial u}{\partial t }(x,T)) =\alpha_1(\frac{\partial u}{\partial t }(x,0) ),x \in \Omega,\\ \end{aligned} \right.$

(1.4)

where $ \alpha_1 $ is the subdifferential of $ j $, i.e., $ \alpha_1 =\partial j,$ and $ j: R \rightarrow R $ is a proper, convex and lower-semi continuous function; $ \alpha_2 : R^+ \bigcup \{0\} \rightarrow R^+ $ is a continuous nonlinear mapping such that $ pt \alpha_2'(t) +(p-1) \alpha_2(t)>0,$ $ 0 \leq \alpha_2(t) \leq k_1 $ for $ t \geq 0 $, $ \lim\limits_{t \rightarrow +\infty}\alpha_2(t) = k_2> 0,$ where $ k_1 $ and $ k_2 $ are positive constants; $ \beta : R \rightarrow R $ is maximal monotone. More details of (1.4) will be presented in Section 2. We shall discuss the existence of solution of (1.4) in $ L^p(0,T;W^{1,p}(\Omega)). $

We may notice that the traditional part $ -\frac{\partial^2u}{\partial t^2} $ in hyperbolic differential equations is replaced by $ -\frac{\partial}{\partial t}(\alpha_1(\frac{\partial u}{\partial t})) $, which leads to the differences in the proofs of the main result. Furthermore, if we set $ j \equiv I,$ $ \alpha_2(t)= 1+ \frac{t}{\sqrt{1+t^{2}}},$ for $ t \geq 0 $, and if $ λ_1\equiv λ_2 \equiv λ,$ then (1.4) becomes to the following hyperbolic capillarity equation with mixed boundaries

$\left\{ \begin{aligned} & - \frac{\partial^2 u}{\partial t^2} -{\rm div}[(1+\frac{|\nabla u|^p}{\sqrt{1+|\nabla u|^{2p}}})|\nabla u|^{p-2}\nabla u]+ λ |u|^{r_1-2}u+λ |u|^{r_2-2}u = f(x,t),\\&(x,t) \in \Omega × (0,T),\\ & -\langle\vartheta,(1+\frac{|\nabla u|^p}{\sqrt{1+|\nabla u|^{2p}}})|\nabla u|^{p-2}\nabla u\rangle \in\beta(u(x,t))-h(x,t),(x,t) \in \Gamma ×(0,T),\\& \gamma u(x,t) = w(x,t),{\rm a.e.} (x,t) \in \Gamma ×(0,T),\\ & u(x,0) = u(x,T),x \in \Omega,\\ & \frac{\partial u}{\partial t }(x,T) = \frac{\partial u}{\partial t}(x,0) ,x \in \Omega. \end{aligned} \right.$

(1.5)

For $ 1 < p \leq 2,$ if we set $ j \equiv I $, $ \alpha_2(t) =(C+t^{\frac{2}{p}})^{\frac{p-2}{2}}t^{\frac{2-p}{p}},$ $ t > 0,C\geq 0 $, and $ λ_2 \equiv 0 $, then (1.4) becomes to the following hyperbolic equation involving generalized $ p $ -Laplacian with mixed boundaries

$\left\{ \begin{aligned} & -\frac{\partial^2 u}{\partial t^2 }-{\rm div} l[(C(x)+|\nabla u|^2) ^{\frac{p-2}{2}}\nabla u]+ λ_1|u|^{r_1-2}u = f(x,t),(x,t) \in \Omega × (0,T),\\& -\langle\vartheta,(C(x)+|\nabla u|^2) ^{\frac{p-2}{2}}\nabla u\rangle \in\beta(u(x,t))-h(x,t),(x,t) \in\Gamma × (0,T),\\& \gamma u(x,t) = w(x,t) {\rm a.e.} (x,t) \in \Gamma × (0,T),\\& u(x,0) = u(x,T),x \in\Omega,\\& \frac{\partial u}{\partial t }(x,T) = \frac{\partial u}{\partial t }(x,0) ,x \in \Omega.\end{aligned} \right.$

(1.6)

If, in (1.6), $ C(x) \equiv 0,$ then (1.6) becomes to the hyperbolic $ p $ -Laplacian boundary value problem.

For $ s \leq 0,$ if we set $ j \equiv I,$ $ \alpha_2(t) =(1+t^{\frac{2}{p}})^{\frac{s}{2}}t^{\frac{m-p+1}{p}},$ $ t > 0,$ $ m\geq 0,m+s+1 = p $, and $ λ_2 \equiv 0 $, then (1.4) becomes to the following hyperbolic curvature equation with mixed boundaries

$\left\{ \begin{aligned} & -\frac{\partial^2 u}{\partial t^2}-{\rm div}[(1+|\nabla u|^2) ^{\frac{s}{2}}|\nabla u|^{m-1}\nabla u] +λ_1|u|^{r_1-2}u = f(x,t),(x,t) \in \Omega × (0,T),\\& -\langle\vartheta,(1+|\nabla u|^2) ^{\frac{s}{2}}|\nabla u|^{m-1}\nabla u\rangle \in \beta(u(x,t))-h(x,t),(x,t) \in\Gamma × (0,T),\\& \gamma u(x,t) = w(x,t) {\rm a.e.} (x,t) \in \Gamma × (0,T),\\& u(x,0) = u(x,T),x \in\Omega,\\& \frac{\partial u}{\partial t }(x,T) = \frac{\partial u}{\partial t }(x,0) ,x \in \Omega. \end{aligned} \right.$

(1.7)

We need the following knowledge to begin our discussion.

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 $ ``\rightarrow" $ and $ ``w-{\rm lim}" $ to denote strong and weak convergence, respectively. Let $ ``X \hookrightarrow Y" $ and $ ``X\hookrightarrow \hookrightarrow Y" $ denote that space $ X $ embedded continuously or compactly in space $ Y,$ respectively. For any subset $ G $ of $ X $, we denote by $ {\rm int} G $ 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 $ {\rm int}G_1 = {\rm 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 $ (see [11, 12]) if $ w-\lim\limits_{t \rightarrow 0}T(x+ty) = Tx $ for any $ x,y \in X. $

A function $ \Phi $ is called a proper convex function on $ X $ (see[11, 12]) if $ \Phi $ is defined from $ X $ to $ (-\infty,+\infty] $, not identically $ +\infty,$ such that

$\Phi((1-λ)x+λ y)\leq (1-λ)\Phi(x)+λ \Phi(y),$

whenever $ x ,y \in X $ and $ 0 \leq λ \leq 1. $ A function $ \Phi: X \rightarrow(-\infty,+\infty] $ is said to be lower-semi continuous on $ X $ (see [11, 12]) if $ \liminf\limits_{y\rightarrow x}\Phi(y)\geq \Phi(x),$ for any $ x \in X $. Given a proper convex function $ \Phi $ 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 $ \Phi $ at $ x,$ and $ \partial \Phi(x) $ is called the subdifferential of $ \Phi $ at $ x $ (see [11]).

Let $ J $ denote the normalized duality mapping form $ X $ into $ 2^{X^*} $ defined by

$Jx = \{f \in X^*: (x,f) =\|x\|\|f\|,\|f\| = \|x\|\},\forall x \in X.$

Since $ X^* $ is strictly convex, $ J $ is single-valued.

A multi-valued operator $ B:X\rightarrow 2^{X^{*}} $ is said to be monotone (see [12]) if its graph $ G(B) $ is a monotone subset of $ X× X^{*} $ in the sense that $ (u_{1}-u_{2},w_{1}-w_{2})\ge0,$ 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})\ge 0 $ 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× X^{*} $ in the sense of inclusion.Also, $ B $ is maximal monotone if and only if $ R(B + λ J) =X^{*},$ for any $ λ> 0. $ The mapping $ B $ is said to be coercive (see [12]) if

$\lim\limits_{n\rightarrow+\infty}{(x_{n},x^*_{n})}/{\|x_{n}\|}=+\infty$

for all $ [x_{n},x^*_{n}]\in G(B) $ such that $ \lim\limits_{n\rightarrow+\infty}\|x_{n}\|= +\infty $.

Let $ B: X \rightarrow 2^{X^{*}} $ be a maximal monotone operator such that $ [0,0] \in G(B),$ then the equation $ J(u_t - u)+ t Bu_t\ni 0 $ 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 the following (see [12]): $ J_t^B u = u_t,B_t u = -\frac{1}{t}J(u_t - u),$ for every $ u \in X $ and $ t> 0. $ Hence, $ [J_t^B u,B_t u]\in G(B). $

Let $ 1 < p < +\infty,$ then $ L^p(0,T ; X) $ denotes the space of all $ X $ -valued strongly measurable functions $ x(t) $ defined a.e. on $ (0,T) $ such that $ \|x(t)\|^p_X $ is Lebesgue integrable over $ (0,T). $ It is well-known that $ L^{p}(0,T; X) $ is a Banach space with the norm defined by

$\|x\|_{L^p(0,T ; X)} = (\displaystyle\int_{0}^T\|x(t)\|_X ^{p}dt)^{\frac{1}{p}}.$

If $ X $ is reflexive, then $ L^p(0,T ; X) $ is reflexive, and its dual space coincides with $ L^{p'}(0,T ; X^*),$ where $ \frac{1}{p}+ \frac{1}{p'}= 1. $ Moreover, $ L^p(0,T ; X) $ is reflexive in the case where $ X $ is reflexive and $ L^p(0,T ; X) $ is strictly (uniformly) convex in the case where $ X $ is strictly (uniformly) convex.

For $ 1 \leq r < p < +\infty,$ if $ X \hookrightarrow Y,$ then $ L^p(0,T ; X)\hookrightarrow L^{r}(0,T ; Y) $.

Lemma 1.1  (see [12]) If $ A: X \rightarrow 2^{X^{*}} $ is a everywhere defined, monotone and hemi-continuous mapping, then $ A $ is maximal monotone.

Lemma 1.2  (see [12]) If $ \Phi : X \rightarrow (-\infty,+\infty] $ is a proper convex and lower-semi continuous function, then $ \partial \Phi $ is maximal monotone from $ X $ to $ X^{*} $.

Lemma 1.3  (see [12]) If $ A_1 $ and $ A_2 $ are two maximal monotone operators in $ X $ such that $ ({\rm int} D(A_1) ) \cap D(A_2) \neq \emptyset,$ then $ A_1 + A_2 $ is maximal monotone.

Theorem 1.1  (see [10]) Let $ X $ be a real reflexive Banach space with both $ X $ and its dual $ X^* $ being strictly convex. Let $ J :X\rightarrow X^* $ be the normalized duality mapping on $ X. $ Let $ A $ and $ B $ be two maximal monotone operators in $ X $. If there exist $ 0 \leq k <1 $ and $ C_1,C_2> 0 $ such that

$(a,J^{-1}(B_t v))\geq -k \|B_t v\|^2 - C_1 \|B_t v\|-C_2,\forall v \in D(A),a \in Av$

and $ t > 0,$ where $ B_t $ is the Yosida approximation of $ B. $ Then $ R(A)+R(B)\simeq R(A+B) $.

Lemma 1.4  (see [13]) Let $ \Omega $ be a bounded conical domain in $ R^N. $ If $ mp > N,$ then $ W^{m,p}(\Omega)\hookrightarrow C_B(\Omega); $ if $ 0 < mp < N $ and $ q = \frac{Np}{N-mp},$ then $ W^{m,p}(\Omega)\hookrightarrow L^q(\Omega); $ if $ mp = N $ and $ p>1,$ then for $ 1 \leq q <+\infty,$ then $ W^{m,p}(\Omega)\hookrightarrow L^q(\Omega). $

Lemma 1.5  (see [13]) Let $ \Omega $ be a domain of $ R^N $ with its boundary $ \Gamma \in C^1,$ then we have the following results

(ⅰ) if $ u \in W^{1,p}(\Omega),$ then the trace $ \gamma u \in W^{1-\frac{1}{p},p}(\Gamma) $ and $ \|\gamma u\|_{W^{1-\frac{1}{p},p}(\Gamma)}\leq K_1 \|u\|_{W^{1,p}(\Omega)}; $

(ⅱ) if $ v \in W^{1-\frac{1}{p},p}(\Gamma),$ then there exists $ u \in W^{1,p}(\Omega) $ such that $ v = \gamma u $ and $ \|u\|_{W^{1,p}(\Omega)} \leq K_2 \|v\|_{W^{1-\frac{1}{p},p}(\Gamma)},$ where $ \gamma : W^{1,p}(\Omega)\rightarrow W^{1-\frac{1}{p},p}(\Gamma) $ denotes the trace operator.

Lemma 1.6  (see [12]) Let $ A: X \rightarrow 2^{X^*} $ be a maximal monotone operator and let $ B: X \rightarrow X^* $ be a hemi-continuous, bounded, coercive and monotone operator with $ D(B) = X,$ then $ R(A+B) = X^*. $

In this paper, unless otherwise stated, we shall assume that $ N\geq 1,2 \leq p<+\infty,1 \leq r_i \leq p $ for $ i = 1,2. $ And $ \frac{1}{p}+\frac{1}{p'}=1,$ $ \frac{1}{r_1}+\frac{1}{r_1'} = 1 $ and $ \frac{1}{r_2}+\frac{1}{r_2'} = 1 $.

In (1.4), $ \Omega $ is a bounded conical domain of a Euclidean space $ R^{N} $ with its boundary $ \Gamma\in C^{1} $ (see [2]), $ T $ is a positive constant, $ λ_1 $ and $ λ_2 $ are non-negative constants, and $ \vartheta $ denotes the exterior normal derivative of $ \Gamma $,

$\gamma :L^p(0,T;W^{1,p}(\Omega))\rightarrow L^p(0,T;W^{1-\frac{1}{p},p}(\Gamma))$

denotes the trace operator. We shall assume that Green's Formula is available.

Suppose that $ \alpha_1 = \partial j $ is continuous, where $ j : R\rightarrow R $ is a proper, convex and lower-semi continuous function. Suppose $ \alpha_2 : R^+ \bigcup \{0\} \rightarrow R^+ $ is a continuous nonlinear mapping such that $ pt \alpha_2'(t) +(p-1) \alpha_2(t)>0,$ $ 0 \leq \alpha_2(t) \leq k_1,$ for $ t \geq0 $, $ \lim\limits_{t \rightarrow +\infty}\alpha_2(t) = k_2 > 0,$ where $ k_1 $ and $ k_2 $ are positive constants. $ \beta : R\rightarrow R $ is maximal monotone such that, for each $ w(x,t) \in L^p(0,T; L^p(\Gamma)),$ $ \beta(w)\in L^p(0,T; L^p(\Gamma)) $.

Now, we present our discussion in the sequel.

Lemma 2.1  (see [14]) For $ u(x,t) \in L^p(0,T;W^{1,p}(\Omega)),$

$\|u\|_{L^p(0,T;W^{1,p}(\Omega))} \leq k_3(\int_0^T\int_{\Omega}|\nabla u|^pdxdt)^{\frac{1}{p}} + k_4,$

where $ k_3 $ and $ k_4 $ are positive constants, $ \frac{2N}{N+1} < p< +\infty $ and $ N \geq 1 $.

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

$\begin{eqnarray*}(w,Bu) &=& \int_0^T\int_{\Omega}\langle \alpha_2(|\nabla u|^p)|\nabla u|^{p-2}\nabla u,\nabla w\rangle dxdt\\&&+λ_1\int_0^T\int_{\Omega}|u|^{r_1-2}uwdxdt+λ_2\int_0^T\int_{\Omega}|u|^{r_2-2}uwdxdt \end{eqnarray*}$

for any $ u,w\in L^p(0,T; W^{1,p}(\Omega)). $ Then $ B $ is everywhere defined, bounded, hemi-continuous, monotone and coercive.

Here $ \langle \cdot,\cdot \rangle $ and $ |\cdot| $ denote the Euclidean inner-product and Euclidean norm in $ R^N. $

Proof  Step 1  $ B $ is everywhere defined. $ \forall u,w \in L^p(0,T; W^{1,p}(\Omega))$,

$\begin{array}{lll}|(w,Bu)| & \leq & \displaystyle\int_0^T\int_{\Omega}k_1|\nabla u|^{p-1}|\nabla w|dxdt +\lambda_1 \displaystyle\int_0^T\int_{\Omega}|u|^{r_1-1}|w|dxdt\\ & & +\lambda_2 \displaystyle\int_0^T\int_{\Omega}|u|^{r_2-1}|w|dxdt\\ & \leq & k_1\|u\|_{L^p(0,T; W^{1,p}(\Omega))}^{\frac{p}{p'}}\|w\|_{L^p(0,T; W^{1,p}(\Omega))}\\ & &+\lambda_1\|w\|_{L^{r_1}(0,T; L^{r_1}(\Omega))} \|u\|_{L^{r_1}(0,T; L^{r_1}(\Omega))}^{\frac{r_1}{r_1'}}\\ & &+ \lambda_2\|w\|_{L^{r_2}(0,T; L^{r_2}(\Omega))} \|u\|_{L^{r_2}(0,T; L^{r_2}(\Omega))}^{\frac{r_2}{r'_2}}.\end{array}$

Since $ W^{1,p}(\Omega)\hookrightarrow L^p(\Omega)\hookrightarrow L^{r_1}(\Omega),$ $ W^{1,p}(\Omega)\hookrightarrow L^p(\Omega)\hookrightarrow L^{r_2}(\Omega),$ then $ \forall v \in W^{1,p}(\Omega),$

$\|v\|_{L^{r_1}(\Omega)} \leq k_5\|v\|_{W^{1,p}(\Omega)},\|v\|_{L^{r_2}(\Omega)} \leq k_6\|v\|_{W^{1,p}(\Omega)},$

where $ k_5 $ and $ k_6 $ are positive constants. Hence,

$\begin{array}{lll} |(w,Bu)| &\leq&k_1\|u\|_{L^p(0,T; W^{1,p}(\Omega))}^{\frac{p}{p'}}\|w\|_{L^p(0,T;W^{1,p}(\Omega))}\\&&+λ_1 k_5 \|u\|_{L^p(0,T;W^{1,p}(\Omega))}^{\frac{r_1}{r_1'}}\|w\|_{L^p(0,T;W^{1,p}(\Omega))}\\&&+ λ_2 k_6 \|u\|_{L^p(0,T;W^{1,p}(\Omega))}^{\frac{r_2}{r_2'}}\|w\|_{L^p(0,T;W^{1,p}(\Omega))},\end{array}$

which implies that $ B $ is everywhere defined. Actually, from the above proof, we know that $ B $ is bounded.

Step 2  $ B $ is strictly monotone. $ \forall u,v \in L^p(0,T;W^{1,p}(\Omega))$,

$\begin{array}{lll}&&(u-v,Bu-Bv) \geq \displaystyle\int_0^T\int_{\Omega}(\alpha_2(|\nabla u|^p)|\nabla u|^{p-1}-\alpha_2(|\nabla v|^p)|\nabla v|^{p-1})(|\nabla u|-|\nabla v|)dxdt \\&&+ λ_1\displaystyle\int_0^T\int_{\Omega}(|u|^{r_1-1}-|v|^{r_1-1})(|u|-|v|)dxdt+λ_2\displaystyle\int_0^T\int_{\Omega}(|u|^{r_2-1}-|v|^{r_2-1})(|u|-|v|)dxdt.\end{array}$

If we set $ f(s) = s^{1-\frac{1}{p}}\alpha_2(s),$ $ s > 0,$ then

$f'(s) = [(1-\frac{1}{p})\alpha_2(s) + s\alpha_2'(s)]s^{-\frac{1}{p}}> 0$

in view of the assumption of $ \alpha_2,$ which implies that $ f $ is strictly monotone. And then $ B $ is strictly monotone.

Step 3  $ B $ is hemi-continuous. In fact, 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) \rightarrow 0,$ as $ t\rightarrow 0 $. Since

$\begin{eqnarray*}&&|(w,B(u+tv)-Bu)|\\&\leq&\int_0^T\int_{\Omega}|\alpha_2(|\nabla u+t\nabla v|^p)|\nabla u+ t\nabla v|^{p-2}(\nabla u + t \nabla v)-\alpha_2(|\nabla u|^p)|\nabla u|^{p-2}\nabla u||\nabla w| dxdt\\&&+λ_1\int_0^T\int_{\Omega}||u+tv|^{r_1-2}(u+tv)-|u|^{r_1-2}u||w|dxdt\\&&+λ_2\int_0^T\int_{\Omega}||u+tv|^{r_2-2}(u+tv)-|u|^{r_2-2}u||w|dxdt\end{eqnarray*}$

by Lebesque's dominated convergence theorem and noticing that $ \alpha_2 $ is continuous, we know that $ \begin{array}{lll}\lim\limits_{t\rightarrow 0}(w,B(u+tv)-Bu) = 0,\end{array} $ and hence $ B $ is hemi-continuous.

Step 4  $ B $ is coercive. For $ u\in L^p(0,T;W^{1,p}(\Omega)),$ let $ \|u\|_{L^p(0,T;W^{1,p}(\Omega))}\rightarrow +\infty. $ Using Lemma 2.1, we find

$\begin{array}{lll}&&\frac{(u,Bu)}{\|u\|_{L^p(0,T;W^{1,p}(\Omega))}}\\&=&\frac{\displaystyle\int_0^T\int_{\Omega}\alpha_2(|\nabla u|^p)|\nabla u|^p dxdt}{\|u\|_{L^p(0,T; W^{1,p}(\Omega))}}+λ_1\frac{\displaystyle\int_0^T\displaystyle\int_{\Omega}|u|^{r_1}dxdt}{\|u\|_{L^p(0,T; W^{1,p}(\Omega))}}+ λ_2\frac{\displaystyle\int_0^T\int_{\Omega}|u|^{r_2}dxdt}{\|u\|_{L^p(0,T; W^{1,p}(\Omega))}}\\ &> & \frac{1}{\|u\|_{L^p(0,T;W^{1,p}(\Omega))}} [k_2\displaystyle\int_0^T\int_{\Omega}|\nabla u|^p dxdt + λ_1\displaystyle\int_0^T\int_{\Omega}|u|^{r_1}dxdt + λ_2\int_0^T\int_{\Omega}|u|^{r_2} dxdt]\\ &> & \frac{1}{\|u\|_{L^p(0,T;W^{1,p}(\Omega))}} k_2\displaystyle\int_0^T\int_{\Omega}|\nabla u|^p dxdt \rightarrow +\infty.\end{array}$

This completes the proof.

Lemma 2.3  Define

$\begin{eqnarray*}S: D(S) &= &\{u(x,t) \in L^{p}(0,T; W^{1,p}(\Omega)): u(x,0) =u(x,T),\alpha_1(\frac{\partial u}{\partial t}(x,0) ) =\alpha_1(\frac{\partial u}{\partial t} (x,T)),\\&&\gamma u = w,\frac{\partial}{\partial t}(\alpha_1(\frac{\partial u}{\partial t}(x,t))) \in L^{p'}(0,T;(W^{1,p}(\Omega))^*)\} \rightarrow (-\infty,+\infty]\end{eqnarray*}$

by

$Su(x,t) = \left\{\begin{array}{l}\displaystyle\int_0^T\int_\Omega j(\frac{\partial u}{\partial t})dxdt,j(\frac{\partial u}{\partial t})\in L^1(0,T;\Omega),\\ +\infty,{\rm otherwise}.\end{array}\right.$

Then the mapping $ S $ is proper, convex and lower-semi continuous.

Proof  It is only need to show that $ S $ is lower-semi continuous on $ L^{p}(0,T; W^{1,p}(\Omega)). $

For this, let $ \{u_n\} $ be such that $ u_n \rightarrow u $ in $ L^{p}(0,T; W^{1,p}(\Omega)) $ as $ n \rightarrow \infty. $ Then there exists a subsequence of $ \{u_{n}\},$ which is still denoted by $ \{u_n\} $ such that

$\frac{\partial u_{n}(x,t)}{\partial t}\rightarrow \frac{\partial u(x,t)}{\partial t}{\rm a.e.} (x,t) \in \Omega × (0,T).$

Since $ j $ is lower-semi continuous, then $ j(\frac{\partial u(x,t)}{\partial t}) \leq\lim\inf\limits_{n \rightarrow \infty }j(\frac{\partial u_{n}(x,t)}{\partial t}) $ a.e. on $ \Omega × (0,T). $ Using Fatou's lemma, we have

$\int_0^T\int_\Omega j(\frac{\partial u(x,t)}{\partial t})dxdt \leq \lim\inf\limits_{n \rightarrow \infty}\int_0^T\int_\Omega j(\frac{\partial u_{n}(x,t)}{\partial t})dxdt.$

Therefore, $ Su \leq \lim\inf\limits_{n \rightarrow\infty}S(u_n),$ whenever $ u_n \rightarrow u $ in $ L^{p}(0,T;W^{1,p}(\Omega)). $ The result follows.

This completes the proof.

Lemma 2.4  Let $ S $ be the same as that in Lemma 2.3. If $ w(x,t) \in \partial S(u(x,t)) $ then

$w(x,t) = -\frac{\partial}{\partial t}(\alpha_1(\frac{\partial u}{\partial t})) {\rm a.e. in} \Omega ×(0,T).$

Proof  Let $ w(x,t) = \frac{\partial \overline{w}(x,t)}{\partial t} $. In view of the definition of subdifferential, we have if $ w(x,t) \in \partial S(u(x,t)),$ then

$\begin{eqnarray}&&\int_0^T\int_\Omega [j(\frac{\partial u}{\partial t})-j(\frac{\partial v}{\partial t})]dxdt\leq \int_0^T\int_\Omega w(x,t)[u(x,t)-v(x,t)]dxdt\nonumber\\&=& \int_0^T\int_\Omega \frac{\partial \overline{w}(x,t)}{\partial t}[u(x,t)-v(x,t)]dxdt = - \int_0^T\int_\Omega\overline{w}(x,t)(\frac{\partial u }{\partial t}-\frac{\partial v}{\partial t})dxdt.\end{eqnarray}$

(2.1)

Let $ E $ be any measurable subset of $ \Omega $ such that for $ t \in (0,T)$,

$ \widetilde{w}(x,t) = \left\{\begin{array}{l} v(x,t),x \in E,\\u(x,t) x \in E^C,\end{array}\right.$

where $ E^C $ is the complement of $ E $ in $ \Omega. $ Taking $ v(x,t)=\widetilde{w}(x,t) $ in (2.1), we have

$\int_0^T\int_E[j(\frac{\partial u}{\partial t})-j(\frac{\partial v}{\partial t})+\overline{w}(x,t)](\frac{\partial u }{\partial t}-\frac{\partial v}{\partial t})dxdt \leq 0.$

In as much as $ E $ was any measurable subset of $ \Omega,$ we have

$j(\frac{\partial u}{\partial t})-j(\frac{\partial v}{\partial t})\leq -\overline{w}(x,t)(\frac{\partial u }{\partial t}-\frac{\partial v}{\partial t}) {\rm a.e.} (x,t) \in \Omega× (0,T).$

Thus

$\overline{w}(x,t)= -\partial j(\frac{\partial u}{\partial t})= - \alpha_1(\frac{\partial u}{\partial t}) {\rm a.e.} (x,t)\in \Omega × (0,T).$

Then

$w(x,t)= -\frac{\partial}{\partial t }(\alpha_1(\frac{\partial u}{\partial t})) {\rm a.e.} (x,t) \in \Omega × (0,T).$

This completes the proof.

Theorem 2.1  For each $ w(x,t) \in L^p(0,T ;W^{1-\frac{1}{p},p}(\Gamma)) $ and $ f(x,t) \in L^{p'}(0,T;(W^{1,p}(\Omega))^*),$ there exists $ u(x,t) \in L^{p}(0,T;W^{1,p}( \Omega)) $ which is the unique solution of the following boundary value problem

$\left\{\begin{array}{ll} -\frac{\partial }{\partial t}(\alpha_1(\frac{\partial u}{\partial t })) - {\rm div}(\alpha_2(|\nabla u|^p)|\nabla u|^{p-2}\nabla u) + λ_1|u|^{r_1-2}u+λ_2|u|^{r_2-2}u = f(x,t),\\ (x,t) \in \Omega × (0,T),\\\gamma u(x,t) = w(x,t) {\rm a.e.} (x,t) \in \Gamma × (0,T),\\u(x,0) = u(x,T),x \in \Omega,\\\alpha_1(\frac{\partial u}{\partial t }(x,T)) =\alpha_1(\frac{\partial u}{\partial t }(x,0) ),x \in \Omega.\end{array}\right. $

(2.2)

In the following of the paper, we denote $ u_{w,f} $ the unique solution of (2.2).

Proof  From Lemmas 1.2, 2.3, 1.6 and 2.2, we know that there exists

$u(x,t) \in L^{p}(0,T; W^{1,p}( \Omega)),$

which satisfies

$\partial S(u(x,t)) + Bu(x,t) = f(x,t). $

(2.3)

Then $ \forall \varphi\in C_0^{\infty}(0,T;\Omega),$

$\begin{eqnarray*}&&\int_0^T\int_{\Omega}-\frac{\partial }{\partial t}(\alpha_1(\frac{\partial u}{\partial t }))\varphi dx dt + \int_0^T\int_{\Omega}\langle \alpha_2(|\nabla u|^p)|\nabla u|^{p-2}\nabla u,\nabla \varphi \rangle dxdt \\&&+\int_0^T\int_{\Omega}λ_1 |u|^{r_1-2} u \varphi dx dt +\int_0^T\int_{\Omega}λ_2 |u|^{r_2-2} u \varphi dx dt =\int_0^T\int_{\Omega}f \varphi dx dt. \end{eqnarray*}$

From the properties of generalized function, we have

$-\frac{\partial }{\partial t}(\alpha_1(\frac{\partial u}{\partial t })) - {\rm div}(\alpha_2(|\nabla u|^p)|\nabla u|^{p-2}\nabla u)+ λ_1|u|^{r_1-2}u+ λ_2|u|^{r_2-2}u = f(x,t) \mbox{a.e. in} \Omega × (0,T).$

Combining with the definition of $ S $, we know that (2.2) has a solution in $ L^p(0,T ;W^{1,p}(\Omega) . $

Uniqueness: let both $ u(x,t) $ and $ v(x,t) $ be solutions of (2.2), then they satisfy (2.3). Thus $ (u - v,Bu - Bv) = - (u - v,\partial S(u) - \partial S(v)) \leq 0,$ since $ \partial S $ is monotone. But $ B $ is monotone too, so $ (u - v,Bu - Bv) = 0,$ which implies that $ u(x,t) = v(x,t) $ since $ B $ is strictly monotone.

This completes the proof.

Lemma 2.5  Define the operator

$A : L^p(0,T ;W^{1-\frac{1}{p},p}(\Gamma)) \rightarrow (L^p(0,T ;W^{1-\frac{1}{p},p}(\Gamma)))^*$

by

$Aw = \langle\vartheta,\alpha_2(|\nabla u_{w,f}|^p)|\nabla u_{w,f}|^{p-2}\nabla u_{w,f}\rangle,\forall w \in L^p(0,T ;W^{1-\frac{1}{p},p}(\Gamma)),$

then $ A $ is maximal monotone on $ L^p(0,T ;W^{1-\frac{1}{p},p}(\Gamma)) $.

Proof Step 1  $ A $ is everywhere defined. $ \forall w_1,w_2 \in L^p(0,T ;W^{1-\frac{1}{p},p}(\Gamma)),$ noticing Lemma 1.5, there exists $ \overline{w_2}\in L^p(0,T ; W^{1,p}(\Omega)) $ such that for $ t\in (0,T),$ $ \gamma \overline{w_2} = w_2 $ and

$\|\overline{w_2}\|_{L^p(0,T ; W^{1,p}(\Omega))}\leq K_2\|w_2\|_{L^p(0,T ; W^{1-\frac{1}{p},p}(\Gamma))}.$

Using Green's formula and (2.2), we have

$\begin{eqnarray*} |(w_2,Aw_1) |&=&|\int_0^T \int_{\Gamma}<\vartheta,\alpha_2(|\nabla u_{w_1,f}|^p)|\nabla u_{w_1,f}|^{p-2}\nabla u_{w_1,f}> w_2d\Gamma(x)dt |\\&=& |\int_0^T \int_{\Omega} \langle\alpha_2(|\nabla u_{w_1,f}|^p)|\nabla u_{w_1,f}|^{p-2}\nabla u_{w_1,f},\nabla \overline{w_2} \rangle dxdt\\&&+ \int_0^T\int_{\Omega}{\rm div}[\alpha_2(|\nabla u_{w_1,f}|^p)|\nabla u_{w_1,f}|^{p-2}\nabla u_{w_1,f}]\overline{w_2}dxdt |\\&\leq&k_1\int_0^T\int_{\Omega}|\nabla u_{w_1,f}|^{p-1}|\nabla\overline{w_2}|dx dt+\int_0^T\int_\Omega |\frac{\partial}{\partial t }(\alpha_1(\frac{\partial u_{w_1,f}}{\partial t}))\overline{w_2}|dxdt\\&&+ \int_0^T\int_{\Omega}|λ_1|u_{w_1,f}|^{r_1-2}u_{w_1,f} + λ_2|u_{w_1,f}|^{r_2-2}u_{w_1,f} - f(x,t)||\overline{w_2}|dxdt\\&\leq& (k_1\|u_{w_1,f}\|_{L^p(0,T ;W^{1,p}(\Omega))}^{\frac{p}{p'}}+ λ_1 \|u_{w_1,f}\|_{L^p(0,T; W^{1,p}(\Omega))}^{\frac{r_1}{r_1'}}\\&&+λ_2\|u_{w_1,f}\|_{L^p(0,T ;W^{1,p}(\Omega))}^{\frac{r_2}{r_2'}}+\|f\|_{L^{p'}(0,T ;(W^{1,p}(\Omega))^*)}\\&&+\|\frac{\partial}{\partial t}\alpha_1(\frac{\partial u_{w_1,f}}{\partial t})\|_{L^{p'}(0,T; (W^{1,p}(\Omega))^*)})\|\overline{w_2}\|_{L^p(0,T ; W^{1,p}(\Omega))},\end{eqnarray*}$

which implies that $ A $ is everywhere defined.

Step 2  $ A $ is monotone. $ \forall w_1,w_2 \in L^p(0,T ;W^{1-\frac{1}{p},p}(\Gamma)) ,$ using Theorem 2.1, there exists

$u_{w_1,f},u_{w_2,f}\in L^p(0,T ; W^{1,p}(\Omega)) $

such that for $ t \in (0,T),$ $ \gamma u_{w_1,f} = w_1 $ and $ \gamma u_{w_2,f}= w_2 $. Using Green's formula, we have the following

\begin{eqnarray*}&&(w_1-w_2,Aw_1- Aw_2) \\&=& \int_0^T \int_{\Gamma}<\vartheta,\alpha_2(|\nabla u_{w_1,f}|^p)|\nabla u_{w_1,f}|^{p-2}\nabla u_{w_1,f}> w_1d\Gamma(x)dt \\&&-\int_0^T \int_{\Gamma}<\vartheta,\alpha_2(|\nabla u_{w_1,f}|^p)|\nabla u_{w_1,f}|^{p-2}\nabla u_{w_1,f}> w_2 d\Gamma(x)dt\\&&-\int_0^T \int_{\Gamma}<\vartheta,\alpha_2(|\nabla u_{w_2,f}|^p)|\nabla u_{w_2,f}|^{p-2}\nabla u_{w_2,f}> w_1d\Gamma(x)dt \\&&+ \int_0^T \int_{\Gamma}<\vartheta,\alpha_2(|\nabla u_{w_2,f}|^p)|\nabla u_{w_2,f}|^{p-2}\nabla u_{w_2,f}> w_2 d\Gamma(x)dt\end{eqnarray*}\begin{eqnarray*}& = &\int_0^T \int_{\Omega}\langle \alpha_2(|\nabla u_{w_1,f}|^p)|\nabla u_{w_1,f}|^{p-2}\nabla u_{w_1,f},\nabla u_{w_1,f}\rangle dxdt \\&&+\int_0^T\int_{\Omega}[-f+λ_1|u_{w_1,f}|^{r_1-2}u_{w_1,f}+λ_2|u_{w_1,f}|^{r_2-2}u_{w_1,f}-\frac{\partial}{\partial t }(\alpha_1(\frac{\partial u_{w_1,f}}{\partial t}))]u_{w_1,f}dxdt\\&&-\int_0^T \int_{\Omega} \langle \alpha_2(|\nabla u_{w_1,f}|^p)|\nabla u_{w_1,f}|^{p-2}\nabla u_{w_1,f},\nabla u_{w_2,f}\rangle dxdt \\&&+\int_0^T\int_{\Omega}[f-λ_1|u_{w_1,f}|^{r_1-2}u_{w_1,f}-λ_2|u_{w_1,f}|^{r_2-2}u_{w_1,f}+\frac{\partial}{\partial t }(\alpha_1(\frac{\partial u_{w_1,f}}{\partial t}))]u_{w_2,f}dxdt\\&&-\int_0^T \int_{\Omega} \langle \alpha_2(|\nabla u_{w_2,f}|^p)|\nabla u_{w_2,f}|^{p-2}\nabla u_{w_2,f},\nabla u_{w_1,f}\rangle dxdt \\&&+\int_0^T\int_{\Omega}[f-λ_1|u_{w_2,f}|^{r_1-2}u_{w_2,f}-λ_2|u_{w_2,f}|^{r_2-2}u_{w_2,f}+\frac{\partial}{\partial t }(\alpha_1(\frac{\partial u_{w_2,f}}{\partial t}))]u_{w_1,f}dxdt\\&&+\int_0^T \int_{\Omega}\langle \alpha_2(|\nabla u_{w_2,f}|^p)|\nabla u_{w_2,f}|^{p-2}\nabla u_{w_2,f},\nabla u_{w_2,f}\rangle dxdt \\&&+\int_0^T\int_{\Omega}[-f+λ_1|u_{w_2,f}|^{r_1-2}u_{w_2,f}+λ_2|u_{w_2,f}|^{r_2-2}u_{w_2,f}-\frac{\partial}{\partial t }(\alpha_1(\frac{\partial u_{w_2,f}}{\partial t}))]u_{w_2,f}dxdt\\&=&(u_{w_1,f} - u_{w_2,f},(B+\partial S)u_{w_1,f} - (B+\partial S)u_{w_2,f}) \geq 0.\end{eqnarray*}

Step 3  $ A $ is hemi-continuous. It suffices to show that for $ w_1 ,w_2,w_3 \in L^p(0,T ; W^{1-\frac{1}{p},p}(\Gamma)) $ and $ k \in [0,1],$

$(w_3,A(w_1+kw_2) - Aw_1) \rightarrow 0$

as $ k \rightarrow 0 . $

In fact, notice again that for $ w_3 \in L^p(0,T ;W^{1-\frac{1}{p},p}(\Gamma)) ,$ there exists $ \overline{w_3} \in L^p(0,T ; W^{1,p}(\Omega)) $ such that $ \gamma \overline{w_3} =w_3 $ for $ t \in (0,T). $

Now we shall compute the following

\begin{eqnarray}&&|(w_3,A(w_1+kw_2) - Aw_1) |\nonumber\\&=& |\int_0^T \int_{\Gamma}\langle\vartheta,\alpha_2(|\nabla u_{w_1+kw_2,f}|^p)|\nabla u_{w_1+kw_2,f}|^{p-2}\nabla u_{w_1+kw_2,f}\rangle w_3 d\Gamma(x)dt \nonumber\\&& -\int_0^T\int_{\Gamma}\langle\vartheta,\alpha_2(|\nabla u_{w_1,f}|^p)|\nabla u_{w_1,f}|^{p-2}\nabla u_{w_1,f}\rangle w_3 d\Gamma(x)dt|\nonumber\\&=& |\int_0^T\int_{\Omega}\langle \alpha_2(|\nabla u_{w_1+kw_2,f}|^p)|\nabla u_{w_1+kw_2,f}|^{p-2}\nabla u_{w_1+kw_2,f},\nabla \overline{w_3}\rangle dx dt \nonumber\\&&+\int_0^T\int_{\Omega}{\rm div}[\alpha_2(|\nabla u_{w_1+kw_2,f}|^p)|\nabla u_{w_1+kw_2,f}|^{p-2}\nabla u_{w_1+kw_2,f}]\overline{w_3}dxdt\nonumber\\&& -\int_0^T\int_{\Omega}\langle \alpha_2(|\nabla u_{w_1,f}|^p)|\nabla u_{w_1,f}|^{p-2}\nabla u_{w_1,f},\nabla \overline{w_3}\rangle dxdt\nonumber\\&&-\int_0^T\int_{\Omega}{\rm div}[\alpha_2(|\nabla u_{w_1,f}|^p)|\nabla u_{w_1,f}|^{p-2}\nabla u_{w_1,f}]\overline{w_3}dxdt| \nonumber\\&=& | \int_0^T\int_{\Omega}\langle \alpha_2(|\nabla u_{w_1+kw_2,f}|^p)|\nabla u_{w_1+kw_2,f}|^{p-2}\nabla u_{w_1+kw_2,f},\nabla\overline{w_3}\rangle dx dt\nonumber\end{eqnarray}\begin{eqnarray} &&-\int_0^T\int_\Omega\frac{\partial}{\partial t}(\alpha_1(\frac{\partial u_{w_1+kw_2,f}}{\partial t})) \overline{w_3}dxdt\nonumber\\&& -\int_0^T\int_\Omega f \overline{w_3}dx dt +\int_0^T \int_\Omega λ_1| u_{w_1+kw_2,f}|^{r_1-2} u_{w_1+kw_2,f}\overline{w_3}dxdt\nonumber\\&&+\int_0^T \int_\Omega λ_2 |u_{w_1+kw_2,f}|^{r_2-2} u_{w_1+kw_2,f}\overline{w_3}dxdt\nonumber\\&&-\int_0^T\int_{\Omega}\langle \alpha_2(|\nabla u_{w_1,f}|^p)|\nabla u_{w_1,f}|^{p-2}\nabla u_{w_1,f},\nabla\overline{w_3}\rangle dx dt + \int_0^T\int_\Omega\frac{\partial}{\partial t}(\alpha_1(\frac{\partial u_{w_1,f}}{\partial t})) \overline{w_3}dxdt\nonumber\\&&+ \int_0^T\int_\Omega f \overline{w_3}dxdt -\int_0^T \int_\Omegaλ_1|u_{w_1,f}|^{r_1-2} u_{w_1,f}\overline{w_3}dx dt-\int_0^T \int_\Omega λ_2|u_{w_1,f}|^{r_2-2}u_{w_1,f}\overline{w_3}dx dt|\nonumber\\&\leq& |(\overline{w_3},Bu_{w_1+kw_2,f}-Bu_{w_1,f})| + \int_0^T\int_\Omega|\alpha_1(\frac{\partial u_{w_1+kw_2,f}} {\partial t})-\alpha_1(\frac{\partial u_{w_1,f}} {\partial t})| |\frac{\partial\overline{w_3}}{\partial t}| dxdt .\end{eqnarray}

(2.4)

Notice that

$\gamma u_{w_1+kw_2,f} = w_1 +kw_2 = \gamma u_{w_1,f}+ k\gamma u_{w_2,f} {\rm a.e. on} \Gamma × (0,T),$

then by using Lemma 1.5, we have

$u_{w_1 + kw_2,f} = u_{w_1,f}+ku_{w_2 ,f} {\rm a.e. in } \Omega × (0,T).$

Thus Lemma2.2, (2.4) and the assumption on $ \alpha_1 $ ensure that

$(w_3,A(w_1+kw_2) - Aw_1) \rightarrow 0,$

as $ k \rightarrow 0,$ which implies that $ A $ is hemi-continuous.

Therefore, Lemma 1.1 implies that $ A $ is maximal monotone. This completes the proof.

Definition 2.1  Define a mapping $ P: L^p(0,T ;W^{1-\frac{1}{p},p}(\Gamma)) \rightarrow R $ by

$\begin{eqnarray*}&&Pw(x,t) = λ_1\int_0^T \int_\Omega |u_{w,f}|^{r_1-2}u_{w,f}dxdt+λ_2\int_0^T \int_\Omega |u_{w,f}|^{r_2-2}u_{w,f}dxdt,\\&&\forall w \in L^p(0,T ;W^{1-\frac{1}{p},p}(\Gamma)),\end{eqnarray*}$

where $ u_{w,f} $ is the unique solution of (2.2).

Theorem 2.2  If

$\frac{1}{T\cdot {\rm meas}(\Gamma)}(\int_0^T\int_\Gamma h(x,t)d\Gamma(x)dt+\int_0^T\int_\Omega f(x,t)dxdt)\in R(\beta+\frac{1}{T\cdot {\rm meas}(\Gamma)}P),$

then nonlinear problem(1.4) has a solution in $ L^p(0,T ; W^{1,p}(\Omega)) $.

Proof  Let $ A $ be defined in Lemma 2.5. Then $ A $ is maximal monotone operator on $ L^p(0,T ; W^{1-\frac{1}{p},p}(\Gamma)) $ or in $ L^{p'}(0,T ; L^{p'}(\Gamma)) $. Define $ Lw = \beta(w),$ where

$D(L) : = \{w \in L^{p'}(0,T; L^{p'}(\Gamma)): \beta(w) \in L^{p}(0,T; L^{p}(\Gamma))\},$

then $ L $ is maximal monotone in $ L^{p'}(0,T; L^{p'}(\Gamma)) $ since $ \beta $ is maximal monotone.

Next, we shall verify that the conditions of Theorem 1.1 are satisfied in $ L^{p'}(0,T; L^{p'}(\Gamma)) $.

$\begin{eqnarray*}&&(Aw,J^{-1} (L_{\mu} (w)))\nonumber\\&=&\int_0^T\int_\Gamma \langle\vartheta,\alpha_2(|\nabla u_{w,f}|^{p})|\nabla u_{w,f}|^{p-2}\nabla u_{w,f}\rangle J^{-1} (\beta_{\mu} (w))d\Gamma(x)dt \nonumber\\&=&\int_0^T\int_\Omega {\rm div}[\alpha_2(|\nabla u_{w,f}|^{p})|\nabla u_{w,f}|^{p-2}\nabla u_{w,f}]\| \beta_{\mu}(u_{w,f})\|_{p}^{2-p}\beta_{\mu}(u_{w,f})|\beta_{\mu}(u_{w,f})|^{p-2}dxdt\nonumber\\&&+ \int_0^T\int_\Omega \langle \alpha_2(|\nabla u_{w,f}|^{p})|\nabla u_{w,f}|^{p-2}\nabla u_{w,f},\nabla J^{-1}(\beta_{\mu} (u_{w,f}))\rangle dxdt\nonumber\\&=&\int_0^T\int_\Omega (-\frac{\partial}{\partial t}(\alpha_1(\frac{\partial u_{w,f}}{\partial t}))+λ_1|u_{w,f}|^{r_1-2}u_{w,f}\\&&+λ_2 |u_{w,f}|^{r_2-2}u_{w,f}-f)\| \beta_{\mu} (u_{w,f})\|_{p}^{2-p}\beta_{\mu}(u_{w,f})|\beta_{\mu}(u_{w,f})|^{p-2} dxdt\\&&+(p-1) \int_0^T\int_\Omega \alpha_2(|\nabla u_{w,f}|^{p})|\nabla u_{w,f}|^{p}\| \beta_{\mu} (u_{w,f})\|_{p}^{2-p}\beta_{\mu}'(u_{w,f})|\beta_{\mu}(u_{w,f})|^{p-2}dxdt\\&\geq& -(\|\frac{\partial}{\partial t}(\alpha_1(\frac{\partial u_{w,f}}{\partial t}))\|_{L^{p'}(0,T;(W^{1,p}(\Omega))^*)}+\|f\|_{L^{p'}(0,T;(W^{1,p}(\Omega))^*)}\\&&+λ_1 \|u_{w,f}\|_{L^{p}(0,T;W^{1,p}(\Omega))}^{\frac{r_1}{r_1'}}\\&&+λ_2\|u_{w,f}\|_{L^{p}(0,T;W^{1,p}(\Omega))}^{\frac{r_2}{r_2'}})\|L_{\mu}(w)\|_{L^p(0,T;L^p(\Gamma))}.\end{eqnarray*}$

Thus from Theorem 1.1, we have $ R(A)+R(L) \simeq R(A+L). $

For $ h(x,t) \in L^{p}(0,T; L^p(\Gamma)),$ we rewrite it as $ h(x,t) = g(x,t) - [g(x,t)-h(x,t)],$ where $ g \in L^{p}(0,T;L^p(\Gamma)). $ For $ g(x,t),$ we write it as $ g(x,t) = g_1(x,t) +g_2(x,t),$ where

$\begin{eqnarray*} g_1(x,t)&= &g(x,t) - \frac{1}{T\cdot {\rm meas}(\Gamma)}\int_0^T\int_\Gamma g(x,t) d\Gamma(x)dt -\frac{1}{T\cdot {\rm meas}(\Gamma)}\int_0^T\int_\Omega f dxdt\\&&+ \frac{1}{T\cdot {\rm meas}(\Gamma)}(λ_1 \int_0^T\int_\Omega|u_{w,f}|^{r_1-2} u_{w,f} dxdt+λ_2 \int_0^T\int_\Omega|u_{w,f}|^{r_2-2} u_{w,f} dxdt)\end{eqnarray*}$

and

$\begin{eqnarray*}g_2(x,t) &=&\frac{1}{T\cdot {\rm meas}(\Gamma)}(\int_0^T\int_\Omega f dxdt+\int_0^T\int_\Gamma h d\Gamma(x)dt\\&&-λ_1\int_0^T\int_\Omega |u_{w,f}|^{r_1-2} u_{w,f} dxdt-λ_2\int_0^T\int_\Omega |u_{w,f}|^{r_2-2} u_{w,f} dxdt)\\&&-\frac{1}{T \cdot {\rm meas}(\Gamma)}\int_0^T\int_\Gamma (h-g)d\Gamma(x)dt.\end{eqnarray*}$

Then

\begin{eqnarray*}&&\int_0^T\int_{\Gamma} g_1(x,t) d\Gamma(x)dt\\&=&-\int_0^T\int_\Omega f dxdt+λ_1 \int_0^T\int_\Omega|u_{w,f}|^{r_1-2} u_{w,f} dxdt+λ_2 \int_0^T\int_\Omega|u_{w,f}|^{r_2-2} u_{w,f} dxdt\\& = & \int_0^T\int_\Omega {\rm div}[\alpha_2(|\nabla u_{w,f}|^{p})|\nabla u_{w,f}|^{p-2}\nabla u_{w,f}] dxdt + \int_0^T \int_\Omega \frac{\partial}{\partial t}\alpha_1(\frac{\partial u_{w,f}}{\partial t})dxdt\end{eqnarray*}\begin{eqnarray*} &=& \int_0^T\int_\Omega {\rm div}[\alpha_2(|\nabla u_{w,f}|^{p})|\nabla u_{w,f}|^{p-2}\nabla u_{w,f}] dxdt\\&=& \int_0^T\int_\Gamma \langle\vartheta,\alpha_2(|\nabla u_{w,f}|^{p})|\nabla u_{w,f}|^{p-2}\nabla u_{w,f}\rangle d\Gamma(x)dt\\& = & \int_0^T\int_\Gamma Aw d\Gamma(x)dt,\end{eqnarray*}

which implies that $ g_1 \in R(A). $

On the other hand, for the small perturbation $ h - g \in L^{p}(0,T; L^p(\Gamma)),$ from the given condition that

$\frac{1}{T \cdot {\rm meas}(\Gamma) }(\int_0^T\int_\Omega f(x,t)dxdt+\int_0^T\int_\Gamma h d\Gamma(x)dt) \in R(\beta+ \frac{1}{T\cdot {\rm meas}(\Gamma)}P),$

we can easily know that $ g_2 \in R(L). $ Therefore, $ g \in R(A+L),$ which implies that $ h \in R(A+L). $ Thus

$-\langle\vartheta,\alpha_2(|\nabla u_{w,f}|^{p})|\nabla u_{w,f}|^{p-2}\nabla u_{w,f}\rangle \in\beta(u_{w,f})- h(x,t).$

That is, (1.4) has a unique solution in $ L^p(0,T ;W^{1,p}(\Omega)) $. This completes the proof.