Singular behavior in nonlinear parabolic equations

Ni, Wei-Ming; Sacks, Paul

doi:10.1090/S0002-9947-1985-0768731-8

Singular behavior in nonlinear parabolic equations
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Trans. Amer. Math. Soc. 287 (1985), 657-671 Request permission

Abstract:

In this paper, we study the well-posedness of the initial-boundary value problems of some quasilinear parabolic equations, namely, nonlinear heat equations and the porous medium equation in the fast-diffusion case. We establish nonuniqueness (local in time) and/or nonregularizing effect of these equations in some critical cases. The key which leads to the resolution of these problems is to study some singular solutions of the elliptic counterparts of these parabolic problems (the so-called $M$-solutions of the Lane-Emden equations in astrophysics).

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