Convergence of the Dirichlet solutions of the very fast diffusion equation |
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Authors: | Kin Ming Hui Sunghoon Kim |
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Institution: | aInstitute of Mathematics, Academia Sinica, Taipei, 10617, Taiwan, ROC |
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Abstract: | For any −1<m<0, positive functions f, g and u0≥0, we prove that under some mild conditions on f, g and u0 as R→∞ the solution uR of the Dirichlet problem ut=(um/m)xx in (−R,R)×(0,∞), u(R,t)=(f(t)|m|R)1/m, u(−R,t)=(g(t)|m|R)1/m for all t>0, u(x,0)=u0(x) in (−R,R), converges uniformly on every compact subset of R×(0,T) to the solution of the equation ut=(um/m)xx in R×(0,T), u(x,0)=u0(x) in R, which satisfies some mass loss formula on (0,T) where T is the maximal time such that the solution u is positive. We also prove that the solution constructed is equal to the solution constructed in Hui (2007) 15] using approximation by solutions of the corresponding Neumann problem in bounded cylindrical domains. |
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Keywords: | MSC: primary 35B40 secondary 35K15 35K65 |
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