## Traveling-wave solutions of parabolic systems with discontinuous nonlinear terms.(English)Zbl 1017.34014

Systems of partial differential equations of the form $$u_t=Au_{xx}+F(u)$$ for $$u:\mathbb{R}^2\rightarrow\mathbb{R}^n$$, $$A$$ a constant $$n\times n$$-matrix and $$F:\mathbb{R}^n\rightarrow\mathbb{R}^n$$, often admit travelling wave solutions (TWS). Such a solution is a vector function of the single variable $$x-ct$$ ($$c$$ a constant velocity) with constant asymptotic behaviours, $$u(+\infty), u(-\infty)$$. In the case that $$n=1$$ and $$F$$ is continuous [A. N. Kolmogorov, Mathematics and its Applications, Soviet Series 25. Dordrecht etc.: Kluwer Academic Publishers (1991; Zbl 0732.01045)] it is completely understood how the existence of TWS is related to the stability of solutions $$u_0$$ to $$F(u_0)=0$$. Partial results are also obtained for the cases where $$n>1$$, e.g., where $$F$$ is continuous and locally monotonic at its zeros.
The present paper establishes an existence theorem for TWS when $$F$$ is discontinuous with isolated singularities $$(F_i^+(u_0)F_i^-(u_0)\leq 0, \forall i, +(-)$$ denoting right(left) limits) and is locally monotonic at such points. If $$F(u(+\infty))=F(u(-\infty))=0$$ the TWS is a strong solution. This theorem is proved under some small technical assumptions concerning $$F$$ by smoothing out its discontinuities, applying results of the authors [Trans. Mosc. Math. Soc. 1990, 59-108 (1990; Zbl 0711.35064)] and then taking a uniform limit to the discontinuous system.

### MSC:

 34B15 Nonlinear boundary value problems for ordinary differential equations 35K55 Nonlinear parabolic equations 35A25 Other special methods applied to PDEs

### Citations:

Zbl 0732.01045; Zbl 0711.35064
Full Text:

### References:

 [1] Berestycki, H.; Larrouturou, B., On the models for planar flames with complex chemistry, C. R. acad. sci. Paris, 317 (Serie 1), 173, (1993) · Zbl 0794.34006 [2] Berestycki, H.; Nicolaenko, B.; Scheurer, B., Travelling wave solutions to combustion models and their singular limits, SIAM J. math. anal., 16, 1207, (1985) · Zbl 0596.76096 [3] Capasso, V., Mathematical structures of epidemic systems, Lecture notes in biomathematics, Vol. 97, (1993), Springer New York · Zbl 0798.92024 [4] Crooks, E.C.M.; Toland, L.F., Traveling waves for reaction-diffusion-convection systems, Topol. methods nonlinear anal., 11, 19, (1998) · Zbl 0920.35075 [5] Fife, P.C.; Mcleod, J.B., The approach of solutions of nonlinear diffusion equations to traveling front solutions, Arch. ration. mech. anal., 65, 335, (1977) · Zbl 0361.35035 [6] Gardner, R., Existence and stability of traveling wave solutions of competition models: a degree theoretic approach, J. differential equations, 44, 343, (1982) · Zbl 0446.35012 [7] Heinze, S., Traveling waves in combustion processes with complex chemical networks, Trans. amer. math. soc., 304, 405, (1987) · Zbl 0661.35042 [8] Kanel’, Ya.I., Stabilization of solutions of the Cauchy problem for the equations arising in the theory of combustion, Mat. sb., 59, 245, (1962) [9] A.N. Kolmogorov, I.G. Petrovsky, N.S. Piskunov, A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem, Bull. Moscow Univ., Math. Mech. 1(6) (1937) 1; in: V.M. Tikhomirov (Ed.), Selected Works of A.N. Kolmogorov, Vol. 1, Kluwer, London, 1991. [10] Lankaster, P., Theory of matrices, (1969), Academic Press New York - London [11] Sard, A., The measure of the critical values of differentiable maps, Bull. amer. math. soc., 48, 883, (1942) · Zbl 0063.06720 [12] Smale, S., An infinite dimensional version of Sard’s theorem, Amer. J. math., 87, 861, (1965) · Zbl 0143.35301 [13] Volpert, V.A.; Volpert, A.I., Application of the Leray-Schauder method to the proof of the existence of wave solutions of parabolic systems, Soviet math. dokl., 37, 138, (1988) · Zbl 0701.35084 [14] Volpert, V.A.; Volpert, A.I., Applications of the rotation theory of vector fields to the study of wave solutions of parabolic equations, Trans. Moscow math. soc., 52, 59, (1990) [15] Volpert, A.I.; Volpert, Vit.A.; Volpert, Vl.A., Traveling wave solutions of parabolic systems, Translations of mathematical monographs, Vol. 140, (1994), American Mathematical Society Providence, RI · Zbl 0835.35048 [16] Zeldovich, Ya.B.; Barenblatt, G.I.; Librovich, V.B.; Makhviladze, G.M., The mathematical theory of combustion and explosion, (1985), Consultants Bureau New York
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