Universal bounds for global solutions of a forced porous medium equation
The author studies the question of obtaining universal a priori estimates for global nonnegative solutions of the problem \[ \begin{aligned} u_t - \Delta\,u^m &= u^p\quad \text{in } \Omega\times (0,T),\\ \left.u\right| _{\partial\Omega} &= 0,\\ \left. u\right| _{t = 0} &= u_0 \end{aligned} \] in a smooth bounded domain \(\Omega\subset\mathbb R^n\), \(n > 1\), with \(1 < m < p\). Here \(u_0\) is a nonnegative continuous function vanishing at \(\partial\Omega\). As a result, the author proves that all global solutions to the problem are uniformly bounded in \(\Omega\times (\tau,\infty)\) by a constant depending on \(\Omega\), \(p\) and \(\tau\) but not on \(u_0\), provided that \(1 < m < p < [(n+1)/(n-1)]m\). Futhermore, it is proven the a priori bound in \(L^{\infty}(\Omega\times (0,\infty))\) depending on \(\| u_0\| _{L^{\infty}(\Omega)}\) under the optimal condition \(1 < m < p < [(n+2)/(n-2)]m\).
scientific article; zbMATH DE number 2102511