On nonextendable solutions and blow-ups of solutions of pseudoparabolic equations with coercive and constant-sign nonlinearities: analytical and numerical study (Q2334908)

The authors study initial-boundary value problem of Cauchy-Dirichlet type for the equations \[ \dfrac{\partial}{\partial t}\big(\Delta u+\lambda u\big)+\Delta u+|u|^{q-1}u=0 \] and \[ \dfrac{\partial}{\partial t}\big(\Delta u+\lambda u\big)+\Delta u+|u|^{q}=0 \] for \(x\in \Omega\) and \(t>0,\) where \(\Omega\subset \mathbb{R}^N,\) \(N=1,2,3,\) is a bounded domain with \(C^{2,\delta}\)-smooth boundary, \(\lambda\) is less than the the first eigenvalue of the Dirichlet problem for the Laplace operator, and \(q>2\) if \(N=1,2,\) while \(2 < q\leq 5\) if \(N=3.\) The solutions are subject to the homogeneous Dirichlet condition \(u|_{\partial\Omega}=0\) \(\forall t>0\) and \(u(x,0)=u_0(x)\) with \(u_0\in H^1_0(\Omega).\) Existence and uniqueness of nonextendable solutions are obtained and sufficient conditions are given for the blow-up of the solutions. The study is supplemented by numerical results that permit to determine the blow-up time.