Blow-up results for nonlinear parabolic equations on manifolds
The author uses a unified approach to derive blow-up results for certain semilinear and quasilinear parabolic equations on manifolds. He first shows that for the equation \[ \begin{cases} \Delta u - \partial_{t}u+ V(x) u^{p} = 0, \\ u(x,0) = u_{0}(x), \quad u \geq 0, \end{cases} \] the critical exponent belongs to the blow-up case on noncompact complete manifolds. He then proves the result for the associated inhomogeneous problem, under some additional assumptions which include nonnegative Ricci curvature, as well as the existence of global positive solutions in the supercritical case. The author next generalizes blow-up results for the homogeneous porous medium equation with nonlinear source \[ \begin{cases} \Delta u^{1+\sigma} - \partial_{t}u + u^{p} = 0, \\ u(x,0) = u_{0}(x),\quad u \geq 0, \end{cases} \] to manifolds, even allowing \(\sigma\) to be somewhat negative. Notably, he does so without making any a priori assumptions on the growth of solutions. Finally, he proves that the inhomogeneous porous medium equation exhibits blow-up behavior in the subcritical case, and under additional conditions, proves the existence of global positive solutions in the supercritical case.
scientific article; zbMATH DE number 1425211
scientific article