The Nonexistence of the Solutions for the Non-Newtonian Filtration Equation with Absorption

Abstract

The paper proves the nonexistence of the solution for the following Cauchy problem
\begin{align*}\begin{cases}u_{t} ={\rm div}\left(\left|\nabla u^{m} \right|^{p-2} \nabla u^{m} \right)-\lambda \; u^{q},&\qquad \left(x,t\right)\in S_{T} ={\mathbb{R}}^N \times \left(0,T\right), \\u\left(x,\; 0\right)=\delta \left(x\right), &\qquad x\in {\mathbb{R}}^N,\end{cases}\end{align*}
under some conditions on $m, p, q, \lambda$, where $\delta $ is Dirac function.