## 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:

 [1] Diaz, J.I.; Kersner, K., MRC report, (1983) [2] S. Irmay, private communication. [3] Kalashnikov, A.S., (), 135-144, (in Russian) [4] Kalashnikov, A.S., U.S.S.R. comput. math. and math. phys., 14, 70-85, (1974) [5] Martinson, L.K.; Pavlov, K.B., U.S.S.R. comput. math. and math. phys., 12, 261-268, (1972) [6] Martinson, L.K., U.S.S.R. comput. math. and math. phys., 16, 141-149, (1976) [7] Kalashnikov, A.S., U.S.S.R. comput. math. and math. phys., 16, 689-696, (1976) [8] Evans, L.C.; Knerr, B.F., Ill. J. math., 23, 153-166, (1979) [9] Kersner, R., Nonlinear anal. theory, (), 1043-1062 [10] M. Bertsch, preprint. [11] Kalashnikov, A.S., Moscow univ. math. bull., 29, 48-53, (1974) [12] Kersner, R.; Kersner, R., Moscow univ. math. bull., Moscow univ. math. bull., 5, 44-51, (1978) [13] Rosenau, P.; Degani, D., Phys. of fluids, 23, 2318-2325, (1980) [14] Gilding, B.H., Arch. rat. mech. anal., 65, 203-222, (1977) [15] Gilding, B.H.; Peletier, L.A., Arch. rat. mech. anal., 6, 127-140, (1976)
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