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

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.


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