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Thermal waves in an absorbing and convecting medium. (English) Zbl 0542.35043

Summary: The quasi-linear parabolic equation \(\partial_ tu=a\partial_{xx}u^{\alpha}+b\partial_ xu^{\beta}-cu^{\gamma}\) exhibits a wide variety of wave phenomena, some of which are studied in this work; and some solvable cases are presented. The motion of the wave front is characterized in terms of \(\alpha\), \(\beta\) and \(\gamma\). Among the interesting phenomena we note the effect of fast absorption (\(b\equiv 0\), \(0<\gamma<1)\) that causes extinction within a finite time, may break the evolving pulse into several sub-pulses and causes the expanding front to reverse its direction. In the convecting case (\(c\equiv 0\), \(b\neq 0)\) propagation has many features in common with Burgers equation, \(\alpha =1\); particularly, if \(0<a\ll 1\), a shock-like transit layer is formed.

MSC:

35K55 Nonlinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
76S05 Flows in porous media; filtration; seepage
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References:

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