Asymptotic behaviour for the heat equation with weak convection and absorption

Article type: Research Article

Authors: Claudi, S. | Guarguaglini, F.R.

Affiliations: Dipartimento di Matematica Pura e Applicata, Università dell’Aquila, L’Aquila, Italy E‐mail: claudi@univaq.it | Dipartimento di Matematica Pura e Applicata, Università dell’Aquila, L’Aquila, Italy E‐mail: guarguaglini@vaxiac.iac.rm.cnr.it

Abstract: We study the large time behaviour of nonnegative solutions of the Cauchy problem \cases{u_t+(\tfrac{u^m}{m})_x = u_{xx} - u^p& ${\rm in}\ {\EBbbR}\times{\EBbbR}^+,$\cr u=u_0 &${\rm in}\ {\EBbbR}\times\{0\},$} where m\Egeq 2, p>1 and u_0 \in L^1({\EBbbR}). We prove that in this range for m the diffusion term prevails over the convection one, thus obtaining the asymptotic profile of the solution in L^q({\EBbbR}) (1\Eleq q\Eleq +\infty).

Journal: Asymptotic Analysis, vol. 16, no. 1, pp. 49-63, 1998