0000000000798083

AUTHOR

Ariel Sánchez

showing 2 related works from this author

Asymptotic behavior for the heat equation in nonhomogeneous media with critical density

2013

Abstract We study the long-time behavior of solutions to the heat equation in nonhomogeneous media with critical singular density | x | − 2 ∂ t u = Δ u , in  R N × ( 0 , ∞ ) in dimensions N ≥ 3 . The asymptotic behavior proves to have some interesting and quite striking properties. We show that there are two completely different asymptotic profiles depending on whether the initial data u 0 vanishes at x = 0 or not. Moreover, in the former the results are true only for radially symmetric solutions, and we provide counterexamples to convergence to symmetric profiles in the general case.

Applied MathematicsMathematical analysisConvergence (routing)Heat equationAnalysisMathematicsCounterexampleNonlinear Analysis: Theory, Methods & Applications
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Large time behavior for a porous medium equation in a nonhomogeneous medium with critical density

2014

Abstract We study the large time behavior of solutions to the Cauchy problem for the porous medium equation in nonhomogeneous media with critical singular density | x | − 2 ∂ t u = Δ u m , in R N × ( 0 , ∞ ) , where m > 1 and N ≥ 3 , with nonnegative initial condition u ( x , 0 ) = u 0 ( x ) ≥ 0 . The asymptotic behavior proves to have some interesting and striking properties. We show that there are different asymptotic profiles for the solutions, depending on whether the continuous initial data u 0 vanishes at x = 0 or not. Moreover, when u 0 ( 0 ) = 0 , we show the convergence towards a peak-type profile presenting a jump discontinuity, coming from an interesting asymptotic simplification…

Conservation lawSingularityApplied MathematicsMathematical analysisConvergence (routing)Initial value problemScale (descriptive set theory)Limit (mathematics)Classification of discontinuitiesPorous mediumAnalysisMathematicsNonlinear Analysis: Theory, Methods & Applications
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