Abstract:

This article studies the critical exponents for the homogeneous evolution $p$-Laplacian equation $u_{t}-\Delta_{p}u=u^{m}+|\nabla u|^{q}$ and its inhomogeneous version $u_{t}-\Delta_{p}u=u^{m}+|\nabla u|^{q}+h(x)$ in $\mathbb{R} ^{N}\times\mathbb{R} ^{+}$, $u(x, 0)=u_{0}(x)$ in $\mathbb{R} ^{N}$, where $N\geq1$, $p$, $m$, $q>1$. We obtain a discontinuous critical exponent result for the homogeneous evolution $p$-Laplacian equation, which demonstrates the gradient term brings about a significant phenomenon of the critical exponent, changing from $m=p-1+p/N$ to $m=\infty$ as $q$ goes to the value $p-N/(N+1)$ from above. Meanwhile, we also investigate the inhomogeneous evolution $p$-Laplacian equation and get a different discontinuous critical exponent result.

Key words: Critical exponents, p-Laplacian equation, Gradient nonlinearity, Blowup, Global existence