Global solutions and blow-up for the wave equation with variable coefficients: II. boundary supercritical source

In this paper, we are concerned with the local and global existence, energy decay rates and finite time blow-up of the solution for the following wave equation

where $Lu=div(A(x)\nabla u)=\sum^{n}_{i,j=1}\frac{\partial}{\partial x_{i}}\bigl{(}a_{ij}(x)\frac{\partial u}{\partial x_{j}}\bigr{)}$, where $A(x)=(a_{ij}(x))$ is a symmetric and positive matrix, and $\frac{\partial u}{\partial\nu_{L}}=\sum^{n}_{i,j=1}a_{ij}(x)\frac{\partial u}{\partial x_{j}}\nu_{i}$, where $\nu=(\nu_{1},\cdots,\nu_{n})$ is the outward unit normal to $\Gamma$. $\Omega$ is a bounded domain of $\mathbb{R}^{n}$($n\geq 3$) with smooth boundary $\Gamma=\Gamma_{0}\cup\Gamma_{1}$. Here, $\Gamma_{0}$ and $\Gamma_{1}$ are closed and disjoint with $meas(\Gamma_{0})\neq 0$.

During the past decades, the problem (1.1) has been widely studied. The condition $h(s)s\leq 0$ means that $h$ represents an attractive force. When $h(s)s\geq 0$ as in the present case, $h$ represents a source term. This situation is more delicate than attractive force, since the solution of (1.1) can blow up. The damping-source interplay in system (1.1) arise naturally in many contexts, for instance, in classical mechanics, fluid dynamics and quantum field theory (cf. [29, 41]). The interaction between two competitive force, that is damping term and source term, make the problem attractive from the mathematical point of view.

For the present case when $h$ is polynomial nonlinear source term such as $h(u)=|u|^{\gamma}u$, the stability of (1.1) has been studied by many authors (see [7, 8, 14, 15, 16, 17, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 31, 36, 37, 38, 39, 43] and a list on references therein), where $h(u)$ is subcritical or critical source. However, very few results addressed wave equations influenced by supercritical sources (cf. [1, 3, 11, 12, 42]). For example, [42] proved the local and global existence, uniqueness and Hadamard well-posedness for the wave equation when source terms can be supercritical or super-supercritical. However, the author do not considered the energy decay and blow-up of the solutions. [11] considered a system of nonlinear wave equations with supercritical sources and damping terms. They proved global existence and exponential and algebraic uniform decay rates of energy, moreover, blow-up result for weak solutions with nonnegative initial energy. But as far as I know, the only problem with considering supercritical source is the constant coefficients case, that is, $A=I$ and dimension $n=3$.

In the case of variable coefficients, that is $A\neq I$, boundary stability of the wave equation was considered in [4, 9, 15, 18, 22]. The wave equations with variable coefficients arise in mathematical modeling of inhomogeneous media in solid mechanics, electromagnetic, fluid flows through porous media. For the variable coefficients problem, the main tool is Riemannian geometry method, which was introduced by [46] and has been widely used in the literature, see [6, 9, 32, 33, 34, 44, 45] and a list of references therein. However, there were very few results considered the source term. For example, [4] proved the uniform decay rate of the energy to the viscoelastic wave equation with variable coefficients and acoustic boundary conditions without damping term. Recently, [26] studied the general decay rate of the energy for the wave equation with variable coefficients and Balakrishnan-Taylor damping and source term without imposing any restrictive growth near zero on the damping term. However, above mentioned examples were not considered Riemannian geometry, and only treated a subcritical source. More recently, [19] proved the uniform energy decay rates of the wave equation with variable coefficients applying the Riemannian geometry method and modified multiplier method. But, it was considered a subcritical source. [23] studied local and global existence, energy decay rate and blow-up of solution for the wave equation with variable coefficients, however it was not considered Reimannian geometry, and was treated the interior supercritical source. There is none, to my knowledge, for the variable coefficients problem having both damping and source terms on Riemannian geometry as well as considering boundary supercritical source.

Our main motivation is constituted by three dimensional case, in which the source term can be supercritical on variable coefficient problem. The differences from previous literatures are as follows:

1. (i)

   Supercritical source for $n\geq 3$.

2. (ii)

   Variable coefficient problem having source term on Riemannian geometry.

3. (iii)

   Blow-up result with positive initial energy as well as nonpositive initial energy.

In order to overcome difficulties to prove above statements, first, we refine the energy space and a constant used in potential well method, because we do not guarantee $H^{1}_{0}(\Omega)\hookrightarrow L^{\gamma+2}(\Gamma_{1})$ since the source term is supercritical. Also we have a hypothesis on damping term for proving existence of solutions and energy decay rates (see Remark 2.2). Second, we use the Faedo-Galerkin method because nonlinear semigroup arguments considered in the previous literatures cannot be used since this paper deal with an operator $-\mu(t)L$, which depends on $t$. Third, we refine the key point constants used to prove blow-up result. So, this paper has improved and generalized previous literatures.

The goal of this paper is to prove the existence result using the Faedo-Galerkin method and truncated approximation method, and classify the energy decay rate applying the method developed in [46] and [35]. Moreover, we prove the blow-up of the weak solution with positive initial energy as well as nonpositive initial energy. This paper is organized as follows: In Section 2, we recall the notation, hypotheses and some necessary preliminaries and introduce our main result. In Section 3, we prove the local existence of weak solutions, and show the global existence of weak solution in each two conditions in Section 4. In Section 5, we prove the uniform decay rate under suitable conditions on the initial data and boundary damping by the differential geometric approach. In Section 6, we prove the blow-up of the weak solution with positive initial energy as well as nonpositive initial energy. by using contradiction method.

We begin this section by introducing some notations and our main results. Throughout this paper, we define the Hilbert space $H^{1}_{0}(\Omega)=\{u\in H^{1}(\Omega);u=0~{}~{}\text{on}~{}~{}\Gamma_{0}\}$ with the norm $||u||_{H^{1}_{0}(\Omega)}=||\nabla u||_{L^{2}(\Omega)}$ and $\mathcal{H}=\{u\in H^{1}_{0}(\Omega);u\in L^{\gamma+2}(\Gamma_{1})\}$ with the norm $||u||_{\mathcal{H}}=||u||_{H^{1}_{0}(\Omega)}+||u||_{L^{\gamma+2}(\Gamma_{1})}$. $||\cdot||_{p}$ and $||\cdot||_{p,\Gamma}$ are denoted by the $L^{p}(\Omega)$ norm and the $L^{p}(\Gamma)$ norm, respectively, and $\langle u,v\rangle=\int_{\Omega}u(x)v(x)dx$ and $\langle u,v\rangle_{\Gamma}=\int_{\Gamma}u(x)v(x)d\Gamma$. Moreover, we need some notations on Riemannian geometry, it is mentioned in [46] and reference therein. For the reader’s comprehension, we will repeat them here.

Let $A(x)=(a_{ij}(x))$ be a symmetric and positive definite matrix for all $x\in\mathbb{R}^{n}$ $(n\geq 3)$ and $a_{ij}(x)$ be smooth functions on $\mathbb{R}^{n}$ satisfying

where $c_{1}$ is a positive constants. Set

For each $x\in\mathbb{R}^{n}$, we define the inner product $g(\cdot,\cdot)=\langle\cdot,\cdot\rangle_{g}$ and the norm $|\cdot|_{g}$ on the tangent space $\mathbb{R}^{n}_{x}=\mathbb{R}^{n}$ by

Then $(\mathbb{R}^{n},g)$ is a Riemannian manifold with Riemann metric $g$. $\nabla_{g}u$ and $D_{g}$ are denoted by the gradient of u and Levi-Civita connection in the Riemannian metric $g$, respectively. It follows that

Let $H$ be a vector field on $(\mathbb{R}^{n},g)$. Then the covariant differential $D_{g}H$ of $H$ determines a bilinear form on $\mathbb{R}^{n}_{x}\times\mathbb{R}^{n}_{x}$, for each $x\in\mathbb{R}^{n}$, by

where $D_{gX}H$ is the covariant derivative of the vector field $H$ with respect to $X$.

$\bf{(H_{1})}$ Hypothesis on $\bf\Omega$.

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded domain, $n\geq 3$, with smooth boundary $\Gamma=\Gamma_{0}\cup\Gamma_{1}$. Here $\Gamma_{0}$ and $\Gamma_{1}$ are closed and disjoint with $meas(\Gamma_{0})\neq 0$. There exists a vector field $H$ on the Riemannian manifold $(\mathbb{R}^{n},g)$ such that

where $\sigma$ is a positive constant and

where $\nu$ represents the unit outward normal vector to $\Gamma$. Moreover we assume that

$\bf{(H_{2})}$ Hypothesis on $\bf\mu$, $\bf f$.

Let $\mu\in W^{1,\infty}(0,\infty)\cap W^{1,1}(0,\infty)$ satisfying following conditions:

where $\mu_{0}$ is a positive constant. We assume that

$\bf(H_{3})$ Hypothesis on $\bf q$.

Let $q:\mathbb{R}\rightarrow\mathbb{R}$ be a nondecreasing $C^{1}$ function such that $q(0)=0$ and suppose that there exist positive constants $c_{3}$, $c_{4}$, and a strictly increasing and odd function $\beta$ of $C^{1}$ class on $[-1,1]$ such that

where $\beta^{-1}$ denotes the inverse function of $\beta$, and $\rho\geq\frac{2(n-2)\gamma-2}{n-(n-2)\gamma}$.

$\bf{(H_{4})}$ Hypothesis on $\bf\gamma$.

Let $\gamma$ be a constant satisfying the following condition:

The energy associated to the problem (1.1) when $h(u)=|u|^{\gamma}u$ is given by

We now state our main results.

This section is devoted to prove the local existence in Theorem 2.1. The proof is based on three steps according to the following condition of the source term $h$:

1. (1)

   Existence of the global solution when $h$ is globally Lipschitz from $H^{1}_{0}(\Omega)$ to $L^{2}(\Gamma_{1})$.

2. (2)

   Existence of the local solution when $h$ is locally Lipschitz from $H^{1}_{0}(\Omega)$ to $L^{2}(\Gamma_{1})$.

3. (3)

   Existence of the local solution when $h$ is locally Lipschitz from $H^{1}_{0}(\Omega)$ to $L^{\frac{\rho+2}{\rho+1}}(\Gamma_{1})$.

Then since the mapping $|u|^{\gamma}u$ is locally Lipschitz from $H^{1}_{0}(\Omega)$ to $L^{\frac{\rho+2}{\rho+1}}(\Gamma_{1})$ (see Remark 2.2), the existence of the local solution can be guaranteed even if $h(u)=|u|^{\gamma}u$.