Abstract
We consider global bounded solutions of fully nonlinear parabolic equations on bounded reflectionally symmetric domains, under nonhomogeneous Dirichlet boundary condition. We assume that, as $t→∞$, the equation is asymptotically symmetric, the boundary condition is asymptotically homogeneous, and the solution is asymptotically strictly positive in the sense that all its limit profiles are strictly positive. Our main theorem states that all the limit profiles are reflectionally symmetric and decreasing on one side of the symmetry hyperplane in the direction perpendicular to the hyperplane. We also illustrate by example that, unlike for equations which are symmetric at all finite times, the result does not hold under a relaxed positivity condition requiring merely that at least one limit profile of the solution be strictly positive.
Highlights
We consider global bounded solutions of fully nonlinear parabolic equations on bounded reflectionally symmetric domains, under nonhomogeneous Dirichlet boundary condition
When considering the asymptotic symmetry of solutions, several natural questions come to mind
Can the asymptotic symmetry be proved if the equation itself is not symmetric, but is merely asymptotically symmetric as t → ∞? one can start thinking about relaxing other conditions: assuming the solutions in question to be asymptotically positive, rather than positive, or replacing the homogeneous Dirichlet boundary condition with an asymptotically homogeneous one
Summary
Ω is a bounded domain in RN satisfying condition (D) from the introduction. The real valued function F is defined on [0, ∞) × Ω × O, where O is an open convex subset of R1+N+N2, invariant under the transformation. R cannot have an arbitrarily fast exponential decay rate It can be proved, that if the nonsymmetric perturbation functions G1, G2 decay to zero with sufficiently fast exponential rate, relative to the Lipschitz constant of the nonlinearity F , Theorem 2.2 holds if the strict positivity assumption (1.7) is relaxed to the weaker assumption requiring the existence of just one positive function in ω(u). This result was mentioned in the survey [20], with reference to the present paper. Since this result is deduced in a standard way from the reflectional symmetry in any direction, we omit the details
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