Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Convergence to nonlinear diffusion waves for solutions of the initial boundary problem to the hyperbolic conservation laws with damping

Authors: Pierangelo Marcati and Ming Mei
Journal: Quart. Appl. Math. 58 (2000), 763-784
MSC: Primary 35L65; Secondary 35A05, 35B40
DOI: https://doi.org/10.1090/qam/1788427
MathSciNet review: MR1788427
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Abstract: In this paper we consider a model of hyperbolic balance laws with damping on the quarter plane $ (x, t) \in {\mathbb{R}_ + } \times {\mathbb{R}_ + }$. By means of a suitable shift function, which will play a key role to overcome the difficulty of large boundary perturbations, we show that the IBVP solutions converge time-asymptotically to the shifted nonlinear diffusion wave solutions of the Cauchy problem to the nonlinear parabolic equation given by the related Darcy's law. We obtain also the time decay rates, which are the optimal ones in the $ {L^{2}}$-sense. Our proof is based on the use of the classical energy method.

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DOI: https://doi.org/10.1090/qam/1788427
Article copyright: © Copyright 2000 American Mathematical Society

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