Simplicity of zeros in scalar parabolic equations. (English) Zbl 0549.35062

Brunovský, P.; Fiedler, B.

J. Differ. Equations 62, 237-241 (1986).

Summary: For \(u_ t=u_{xx}+f(x,u)\) with Dirichlet boundary-condition the following result is obtained: if \(u(t,1)\) has only finitely many sign changes at \(t=0\), then \(u(t,\cdot)\) has only simple zeros for a set of \(t>0\) which is open and dense in the maximal interval of existence of the solution \(u(t,\cdot)\). This result is useful for a study of orbits connecting critical points of the above equation. Its proof relies on maximum principle arguments.

MSC:

35K55	Nonlinear parabolic equations
35B05	Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B50	Maximum principles in context of PDEs

Keywords:

reaction-diffusion equation; Dirichlet boundary-condition; sign changes; simple zeros; maximal interval of existence; orbits connecting critical points; maximum principle

References:

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