Exponential energy decay of solutions for an integro-differential equation with strong damping
Consider \(\Omega\subset\mathbb R^N\), \(N\geq 1\), a bounded domain with a smooth boundary \(\partial\Omega\) so that the divergence theorem can be applied, the function \(M\in C^1([0,+\infty))\), \[ M(s)= m_0+ bs^\gamma \] with \(m_0\geq 0\), \(b\geq 0\), \(\gamma\geq 1\), a nondecreasing function \(g\in C^1([0,+\infty))\) so that \[ (\forall s\geq 0)(g(s)\geq 0),\quad m_0- \int^{+\infty}_0 g(s)\,ds= 1>0, \] a nonlinear function \(f(u)= |u|^{p-2}u\), \(p> 2\), where \(u: \Omega\times [0,+\infty)\to \mathbb R\). The author studies the initial boundary value problem for the integro-differential equation \[ \begin{gathered} u_{tt}- M(\|\nabla u\|^2_2)\Delta u+\int^t_0 g(t- s)\Delta u(s)\,ds-\Delta u_t= f(u),\\ u(x,0)= u_0(x),\quad u_t(x,0)= u_1(x),\quad x\in\Omega,\\ u(x,t)= 0,\quad x\in\partial\Omega,\quad t> 0.\end{gathered}\tag{1} \] Using the Sobolev space \(W^{m,p}(\Omega)\) and \(H^1_0(\Omega)= \overline{C^\infty_0(\Omega)}\) with respect to the norm \[ \| u\|_{H^1_0}=\|\nabla u\|_2, \] the author proves that if \(u_0\in H^1_0(\Omega)\cap W^{2,2}(\Omega)\), \(u_1\in L^2(\Omega)\), then there exists a unique solution \(u\) of the problem (1) satisfying \[ u\in C([0,T], H^1_0(\Omega)\cap W^{2,2}(\Omega)), \] \[ u_t\in C([0, T], L^2(\Omega)\cap L^2([0,T], H^1_0(\Omega))). \]
scientific article; zbMATH DE number 5673245
scientific article