On the existence and uniqueness of time periodic solutions for a semilinear heat equation in the whole space. (English) Zbl 1357.35021
Summary: This paper is concerned with the existence and uniqueness of time periodic solutions in the whole-space \(\mathbb{R}^N\) for a heat equation with nonlinear term. The nonlinear term we considered is of this type, \(|u|^{q-1}u+f(x,t)\), with \(q>\frac N{N-2}\), \(N>2\). We show that there exists a unique time periodic solution when the source term \(f\) is small. In fact, \(\frac N{N-2}\) is a critical exponent; when \(1<q\leq\frac N{N-2}\), there is no time periodic solution.
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