De Pablo et al. [Proc. Roy. Soc. Edinburgh Sect. A 138 (2008), 513-530] considered a nonlinear boundary value problem for a porous medium equation with a convection term, and they classified exponents of nonlinearities which lead either to the global-in-time existence of solutions or to a blow-up of solutions. In their analysis they left open the case of a certain critical range of exponents. The purpose of this note is to fill this gap.
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We consider the existence of positive solutions of the equation $1/λ(t) (λ(t)φ_p(x'(t)))' + μf(t,x(t),x'(t)) =0$, where $φ_p(s) = |s|^{p-2}s$, p > 1, subject to some singular Sturm-Liouville boundary conditions. Using the Krasnosel'skiĭ fixed point theorem for operators on cones, we prove the existence of positive solutions under some structure conditions.
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This paper is mainly concerned with the blow-up and global existence profile for the Cauchy problem of a class of fully nonlinear degenerate parabolic equations with reaction sources.
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