Well-posedness of a semilinear heat equation with weak initial data

Abstract

This article mainly consists of two parts. In the first part the initial value problem (IVP) of the semilinear heat equation

$$\begin{gathered} \partial _t u - \Delta u = \left| u \right|^{k - 1} u, on \mathbb{R}^n x(0,\infty ), k \geqslant 2 \hfill \\ u(x,0) = u_0 (x), x \in \mathbb{R}^n \hfill \\ \end{gathered} $$

with initial data in$\dot L_{r,p}$ is studied. We prove the well-posedness when

$$1< p< \infty , \frac{2}{{k(k - 1)}}< \frac{n}{p} \leqslant \frac{2}{{k - 1}}, and r =< \frac{n}{p} - \frac{2}{{k - 1}}( \leqslant 0)$$

and construct non-unique solutions for

$$1< p< \frac{{n(k - 1)}}{2}< k + 1, and r< \frac{n}{p} - \frac{2}{{k - 1}}.$$

In the second part the well-posedness of the avove IVP for k=2 with μ₀ɛH ^s(ℝⁿ) is proved if

$$ - 1< s, for n = 1, \frac{n}{2} - 2< s, for n \geqslant 2.$$

and this result is then extended for more general nonlinear terms and initial data. By taking special values of r, p, s, and u₀, these well-posedness results reduce to some of those previously obtained by other authors [4, 14].