## 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.

### 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
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### References:

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