Critical exponent for the bipolar blowup in a semilinear parabolic equation
The blowup of solutions of the initial-boundary value problem: \[ u_t=\Delta u+| u|^{p-1}u\quad\text{in }\Omega\times(0,\infty) \] \[ u(x,t)=0\quad \text{on }\partial\Omega\times (0,\infty),\qquad u(x,0)=u_0(x) \quad\text{in }\Omega \] is studied, where \(\Omega\) is a cone in \(\mathbb{R}^N\). A solution exhibits bipolar blowup if \[ \sup_{x\in \Omega}u(x,t) \to+\infty \quad \text{and}\quad \inf_{x\in\Omega} u(x,t) \to-\infty \] for some \(t\to\infty\). The authors show that there exists a critical exponent \(p^x>1\) for the bipolar blowup in the following sense: if \(1<p\leq p^x\), then there exist arbitrarily small initial data such that the solution exhibits the bipolar blowup, whereas if \(p>p^*\), then the bipolar blowup does not occur for any sufficiently small initial data. The value of \(p^*\) is expressed in terms of the dimension \(N\) and the second Dirichlet eigenvalue of the Laplace-Beltrami operator on \(\Omega \times S^{N-1}\). In the case of \(\Omega=R\) \(k\)-bipolar blowup is also defined, and critical exponents for the \(k\)-polar blowup are determinated.
scientific article; zbMATH DE number 1130513
scientific article