Instantaneous shrinking of the support of a solution of the Cauchy problem for a quasilinear heat equation
This paper concerns with Cauchy problem NEWLINE\[NEWLINE u_t=\Delta u+ u^\lambda, \quad (x,t)\in \mathbb R^N\times (0,T), NEWLINE\]NEWLINE NEWLINE\[NEWLINE u(x,0)=u_0(x), \quad x\in \mathbb R^N,NEWLINE\]NEWLINE in the case, when \(u_0\) is locally bounded Radon measure. Let us denote \(B_t(x)=\{ y\in \mathbb R^N: | y-x| \leq t^{1/2}\}\), NEWLINE\[NEWLINE \phi(t)=\frac{1}{| B_t(x)| }\int_{B_t(x)}u_0(x)\,dx, NEWLINE\]NEWLINE \(\phi_t(r)=\sup\limits_{| x| =r}\phi_t(x)\), \(r>0\). The author proves that instantaneous support shrinking takes place if and only if \(\phi_t(r)\to 0\), \(r\to \infty\) and gives sharp two-sided estimates of support.
scientific article; zbMATH DE number 6062203