Existence and nonexistence of local/global solutions for a nonhomogeneous heat equation. (English) Zbl 1284.35071
Summary: We study the existence of local/global solutions to the Cauchy problem
\[
\begin{cases} \rho(x)u_t=\Delta u+q(x)u^p, \quad & (x,t)\in \mathbb R^N \times (0,T),\\ u(x,0)=u_{0}(x)\geq 0, & x \in \mathbb R^N \end{cases}
\]
with \(p > 0\) and \(N\geq 3\). We describe the sharp decay conditions on \(\rho\), \(q\) and the data \(u_0\) at infinity that guarantee the local/global existence of nonnegative solutions.
MSC:
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35K55 | Nonlinear parabolic equations |
| 35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
| 35Q35 | PDEs in connection with fluid mechanics |
| 92C17 | Cell movement (chemotaxis, etc.) |
| 35B44 | Blow-up in context of PDEs |
| 35K15 | Initial value problems for second-order parabolic equations |
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