Uniqueness and stability of regional blow-up in a porous-medium equation |
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Authors: | Carmen Cort zar, Manuel del Pino,Manuel Elgueta |
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Affiliation: | a Departamento de Matemáticas, Pontificia Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile;b Departamento de Ingeniería Matemática, Universidad de Chile, and Centro de Modelamiento Matemático, UMR2071 CNRS-UChile, Casilla 170, Correo 3, Santiago, Chile |
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Abstract: | We study the blow-up phenomenon for the porous-medium equation in RN, N 1, ut=Δum+um, m>1, for nonnegative, compactly supported initial data. A solution u(x,t) to this problem blows-up at a finite time . Our main result asserts that there is a finite number of points x1,…,xk RN, with |xi−xj| 2R* for i≠j, such that Here w*(|x|) is the unique nontrivial, nonnegative compactly supported, radially symmetric solution of the equation in RN and R* is the radius of its support. Moreover u(x,t) remains uniformly bounded up to its blow-up time on compact subsets of . The question becomes reduced to that of proving that the ω-limit set in the problem consists of a single point when its initial condition is nonnegative and compactly supported. |
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Keywords: | Mathematical subject codes: 35K65 35B40 |
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