Blow-up on the boundary for the heat equation
Consider the heat equation \(\partial_t u=\Delta u\) in \(\Omega\times(0,T)\) complemented by the nonlinear boundary condition \(\partial u/\partial\nu=f(u)\) and the initial condition \(u(\cdot,0)=u_0\). Here \(\Omega\) is a bounded domain in \(\mathbb R^n\) with a Lipschitz boundary \(S\), and the function \(f\) satisfies the growth condition \(|f(u)|\leq C(|u|^p+1)\), \(p>1\). It is known (for \(f\) sufficiently smooth) that the boundedness of the trace of the solution \(u(\cdot,t)\) in \(L^q(S)\) guarantees an \(L^\infty\) bound for the solution provided \(q>(n-1)(p-1)\). Similarly, for \(f\) and \(S\) smooth enough, the boundedness of \(u(\cdot,t)\) in \(L^r(\Omega)\) implies an \(L^\infty\) bound provided \(r>n/(p-1)\). The authors construct an example of a blowing up solution indicating that the conditions on \(q\) and \(r\) are optimal (up to the equality sign). More precisely, they have \(n=2\), \(\Omega=(0,1)^2\), \(f(u)=u^p\) on \(S_0:=\{(x,y)\in S:xy=0\}\), \(f(u)=-u^p\) on \(S\setminus S_0\), \(u_0(x,y)=[(p-1)(x+y+\rho)]^{-1/(p-1)}\), \(\rho>0\), and the solution stays bounded in \(L^r(\Omega)\) and \(L^q(S)\) for any \(r<2/(p-1)\) and \(q<1/(p-1)\).
scientific article; zbMATH DE number 1406317