An optimal Liouville-type theorem for radial entire solutions of the porous medium equation with source
The author considers nonnegative weak solutions of the porous medium equation with source. Assume \(m>1\) and \(1 < p/m < p_s\), where \(p_s\) is the Sobolev exponent defined by \(p_s = (n+1)/(n-2)\) if \(n >2\) and \(\infty\) if \(n \leq 2\). The author proves that the equation \[ u_t - \Delta u^m = u^p,\quad x \in {\mathbb R}^n, t \in {\mathbb R} \] has no nontrivial, bounded radial solutions \(u \geq 0\). In one space-dimensional, the conclusion of the result mentioned above remains true without the assumption of radial symmetry. In fact the conclusions of the results remain true without the boundedness assumption on \(u\). The proof is based on intersection-comparison arguments and a key step is to show the positivity of possible bounded radial entire solutions. As an auxiliary result, the author establishes pointwise gradient estimates for bounded, radial nonnegative solutions of the equation \(u_t - \Delta u^m = u^p\), which may be of independent interest.
scientific article; zbMATH DE number 5555288
scientific article