Discontinuous critical Fujita exponents for the heat equation with combined nonlinearities
Mohamed Jleli, Bessem Samet, Philippe Souplet
TL;DR
This work analyzes blow-up versus global existence for the nonlinear heat equation with a gradient nonlinearity and its inhomogeneous counterpart. Using a combination of test-function arguments, scaling techniques, and eigenfunction-based approaches, it derives sharp Fujita-type thresholds for both positive and sign-changing data. A key finding is a discontinuous jump of the critical exponent as the gradient exponent crosses $1+\frac{1}{n+1}$ (and analogous thresholds in the inhomogeneous case with $p$ and $q$ relative to $\frac{n}{n-2}$ and $\frac{n}{n-1}$). These results illuminate how the gradient term can alter Fujita-type behavior and extend classical results to problems with combined nonlinearities, with implications for diffusion-transport models.
Abstract
We consider the nonlinear heat equation $u_t-Δu =|u|^p+b |\nabla u|^q$ in $(0,\infty)\times \R^n$, where $n\geq 1$, $p>1$, $q\geq 1$ and $b>0$. First, we focus our attention on positive solutions and obtain an optimal Fujita-type result: any positive solution blows up in finite time if $p\leq 1+\frac{2}{n}$ or $q\leq 1+\frac{1}{n+1}$, while global classical positive solutions exist for suitably small initial data when $p>1+\frac{2}{n}$ and $q> 1+\frac{1}{n+1}$. Although finite time blow-up cannot be produced by the gradient term alone and should be considered as an effect of the source term $|u|^p$, this result shows that the gradient term induces an interesting phenomenon of discontinuity of the critical Fujita exponent, jumping from $p=1+\frac{2}{n}$ to $p=\infty$ as $q$ reaches the value $1+\frac{1}{n+1}$ from above. Next, we investigate the case of sign-changing solutions and show that if $p\le 1+\frac{2}{n}$ or $0<(q-1)(np-1)\le 1$, then the solution blows up in finite time for any nontrivial initial data with nonnegative mean. Finally, a Fujita-type result, with a different critical exponent, is % also obtained for sign-changing solutions to the inhomogeneous version of this problem.