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Blow-up results for nonlinear parabolic equations on manifolds. (English) Zbl 0954.35029

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.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
58J35 Heat and other parabolic equation methods for PDEs on manifolds
35K55 Nonlinear parabolic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B33 Critical exponents in context of PDEs
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