Existence and a general decay result for a plate equation with nonlinear damping and a logarithmic source term
The authors deal with the existence and decay of solutions of the nonlinear plate problem \[ \begin{aligned} & u_{tt}+\Delta^2u+u+ h(u_t) = ku\ln|u|,\;x\in \Omega\subset \mathbb{R}^2,\;t>0; \\ & u(x,t)=\frac{\partial u}{\partial \nu}(x,t)=0,\;x\in \partial \Omega,\;t>0;\quad u(x,0)=u_0(x),\;u_t(x,0)=u_1(x),\;x\in \Omega. \end{aligned} \] Using the Galerkin method and the properties of \(h\) and \(k\) they establish for \((u_0,u_1)\in H_0^2(\Omega)\times L^2(\Omega)\) the existence of a weak solution \(u\in C([0,T],H_0^2(\Omega))\cap C^1([0,T],L^2(\Omega))\cap C^2([0,T],H^{-2}(\Omega)),\;T>0\). The decay of the energy functional \[ E(t)=\frac 12 \left(\|u_t\|_2^2+\|\Delta u\|^2_2+\|u\|_2-\int_\Omega u^2\ln|u|^k\,\text{d}x\right)+\frac k4\|u\|_2^2 \] is verified.
scientific article; zbMATH DE number 6861529