Abstract
In this paper we consider the Cauchy problem for the heat equation with a nonnegative potential decaying quadratically at the space infinity and investigate local concavity properties of the solution. In particular, we give a sufficient condition for the solution to be quasi-concave in a ball for any sufficiently large t, and discuss the optimality of the sufficient condition, identifying a threshold for the occurrence of local quasi-concavity.
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Andreucci, D., Ishige, K. Local quasi-concavity of the solutions of the heat equation with a nonnegative potential. Annali di Matematica 192, 329–348 (2013). https://doi.org/10.1007/s10231-011-0226-x
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DOI: https://doi.org/10.1007/s10231-011-0226-x
Keywords
- Concavity of solutions of parabolic equations
- Local quasi-concavity for large times
- Heat equation with potential
- Hot spots