On nonlinear parabolic equations of very fast diffusion
The authors are concerned with the weak solvability of \[ \partial_t u=\nabla\cdot(u^{m-1}\nabla u)\quad\text{in }{\mathbb{R}}^n\times (0,T),\qquad u(x,0)=f(x),\quad x\in {\mathbb{R}}^n, \tag{1} \] where \(n\in\mathbb{N}\), \(T>0\), \(m<0\) and \(f\in L^1_{\text{loc}}({\mathbb{R}}^n;[0,\infty))\). \textit{J. L. Vázquez} [J. Math. Pures Appl., IX. Sér. 71, No. 6, 503-526 (1992; Zbl 0831.35082)] has established that the solvability of the initial value problem implies \(f\not\in L^1({\mathbb{R}}^n )\), and the authors showed in an earlier paper [Indiana Univ. Math. J. 43, No. 2, 703-728, (1994; Zbl 0806.35086)] that \(\displaystyle \limsup_{R\to\infty}{1\over{R^{n-2/(1-m)}}}\int_{B_R} f\geq CT^{1/(1-m)}\) is necessary for (1) to be solvable on \((0,T)\), where \(C\) is a constant depending only on \(n\) and \(m\), and \(B_R\) is the ball with radius \(R\) and center \(0\) in \({\mathbb{R}}^n\). In this paper they obtain nonexistence provided that \[ \limsup_{R\to\infty}R^{2m/(1-m)}\int_0^R\Bigl[{{ds} \over{\omega_ns^{n-1}}}\int_{B_s}f\Bigr]<C_1 T^{1/(1-m)}\tag{2} \] with \(C_1\) an explicit constant depending only of \(n\) and \(m\) and \(\omega_n\) the surface measure of the unit sphere. The main theorem of the paper is an existence result which guarantees solvability of (1) for example in the special case when \(f\) is radially symmetric and locally bounded and \(<\) in (2) is replaced by \(\geq\). Such an assertion cannot be expected for general \(f\) as an example reveals.
scientific article; zbMATH DE number 1042621