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Large time behavior for convection-diffusion equations in \(\mathbb{R}{} ^ N\). (English) Zbl 0762.35011

Summary: We describe the large time behavior of solutions of the convection- diffusion equation \[ u_ t-\Delta u=a\cdot\nabla(| u|^{q- 1}u)\quad\text{in } (0,\infty)\times\mathbb{R}^ N \] with \(a\in\mathbb{R}^ N\) and \(q\geq 1+1/N\), \(N\geq 1\). When \(q=1+1/N\), we prove that the large time behavior of solutions with initial data in \(L^ 1(\mathbb{R}^ N)\) is given by a uniparametric family of self-similar solutions. The relevant parameter is the mass of the solution that is conserved for all \(t\). Our result extends to dimensions \(N>1\) well known results on the large time behavior of solutions for viscous Burgers equations in one space dimension. The proof is based on La Salle’s invariance principle applied to the equation written in its self-similarity variables.
When \(q>1+1/N\) the convection term is too weak and the large time behavior is given by the heat kernel. In this case, the result is easily proved applying standard estimates of the heat kernel on the integral equation related to the problem.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35K05 Heat equation
35K15 Initial value problems for second-order parabolic equations
76R99 Diffusion and convection
35Q35 PDEs in connection with fluid mechanics
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