Discontinuous critical Fujita exponents for the heat equation with combined nonlinearities

We consider the nonlinear heat equation $u_t-\mathrm{\Delta }u=\mathrm{\mid }u{\mathrm{\mid }}^p+b\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^q$ in $(0,\mathrm{\infty })\times {\mathbb{R}}^n$, where $n\ge 1$, $p>1$, $q\ge 1$ 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 $p\le 1+\frac{2}{n}$ or $q\le 1+\frac{1}{n+1}$, 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 $\mathrm{\mid }u{\mathrm{\mid }}^p$, 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 $p=\mathrm{\infty }$ 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.