An optimal Liouville-Type Theorem for Radial Entire Solutions of the Porous Medium Equation with Source<br/>

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An optimal Liouville-Type Theorem for Radial Entire Solutions of the Porous Medium Equation with Source

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 40807 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Hesaraki, Mahmoud
Abstract:
In this thesis , we consider nonnegative (continuous) weak solutions of the porous medium equation with source , u_t-∆u^m=u^p, and p>m>1 .
Assume that, m>1 and 1< p/m u_t-∆u^m=u^p,xϵR^n ,tϵR
has no nontrivial, bounded radial solutions u≥0 .
In one space-dimensional, the conclusion of the result mentioned above remains true without the assumption of the radial symmetry. The proof is based on the intersection-comparison arguments , zero number argum- ents and a key step is to show the positivity of possible bounded radial entire solutions . As an auxiliary result, we established pointwise gradient estimates for bounded, radial nonnegative solutions of the equation u_t-∆u^m=u^p .
Keywords:
Parabolic Equations ; Parabolic Liouville-Type Theorem ; Bounded Radial Solutions ; Porous Medium Equation ; Bernstein-Type Estimates

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