Exponential growth for a fractionally damped wave equation
Summary: This paper is concerned with the fractional differential problem \[ \begin{cases} u_{tt}+ \partial_t^{1+\alpha} u=\Delta u+|u |^{p-1}u \quad & (x\in\Omega, t>0)\\ u(x,t)=0 \quad & (x\in\Gamma, t>0)\\ u(x,0)= u_0(x),u_t(x,0)= u_1(x)\quad & (x\in\Omega) \end{cases} \] where \(p>1\), \(-1< \alpha<1\), \(u_0\) and \(u_1\) are given functions, \(\Omega\) is a bounded domain of \(\mathbb{R}^N\) with smooth boundary \(\Gamma\), and \(\partial_t^{1+\alpha}\) is Caputo's fractional derivative of order \(1+\alpha\). It is proved that the solution is unbounded and grows up exponentially in the \(L_p\)-norm for sufficiently large initial data. To this end we use some techniques based on Fourier transforms and some inequalities such as the Hardy-Littlewood inequality.
scientific article; zbMATH DE number 1895119
scientific article