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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
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