Discontinuous critical Fujita exponents for the heat equation with combined nonlinearities
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- by Mohamed Jleli, Bessem Samet and Philippe Souplet PDF
- Proc. Amer. Math. Soc. 148 (2020), 2579-2593 Request permission
Abstract:
We consider the nonlinear heat equation $u_t-\Delta u =|u|^p+b |\nabla u|^q$ in $(0,\infty )\times \mathbb {R}^n$, 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 obtained for sign-changing solutions to the inhomogeneous version of this problem.References
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Additional Information
- Mohamed Jleli
- Affiliation: Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia
- MR Author ID: 777258
- Email: jleli@ksu.edu.sa
- Bessem Samet
- Affiliation: Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia
- MR Author ID: 719011
- Email: bsamet@ksu.edu.sa
- Philippe Souplet
- Affiliation: Université Sorbonne Paris Nord, CNRS UMR 7539, Laboratoire Analyse, Géométrie et Applications, 93430 Villetaneuse, France
- MR Author ID: 314071
- Email: souplet@math.univ-paris13.fr
- Received by editor(s): July 10, 2019
- Received by editor(s) in revised form: October 24, 2019
- Published electronically: February 4, 2020
- Additional Notes: The first author was supported by Researchers Supporting Project number (RSP-2019/57), King Saud University, Riyadh, Saudi Arabia
- Communicated by: Ryan Hynd
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 2579-2593
- MSC (2010): Primary 35K05, 35B44, 35B33
- DOI: https://doi.org/10.1090/proc/14953
- MathSciNet review: 4080898