Quantitative unique continuation for the semilinear heat equation in a convex domain
Let \(\Omega\) be a bounded convex smooth domain in \(\mathbb R^d\). The authors study unique continuation properties of solutions of the semilinear heat equation \(u_t-\Delta u=g(u)\) in \(\Omega\times(0,T_*)\) complemented by the homogeneous Dirichlet boundary conditions. Assume first that \(g\) is \(C^1\) and \(g'\) satisfies suitable growth conditions guaranteeing the well-posedness of the problem in \(L^2(\Omega)\). Let \(u,v\) be two solutions with initial data in \(L^2(\Omega)\), let \(T>0\) be less than the minimum of their maximal existence times and \(\omega\subset\Omega\) be open and non-empty. One of the main results of the paper guarantees the estimate \[ \int_\Omega|(u-v)(x,T)|^2\,dx \leq C\biggl(\int_\omega|(u-v)(x,T)|^2\,dx\biggr)^\alpha, \] where \(\alpha\in(0,1)\) depends on \(\Omega\) and \(\omega\) and \(C>0\) depends on \(\Omega,\omega,T\), the norms of the initial data in \(L^2(\Omega)\) and the growth of \(g'\). In addition, if \(u(\cdot,T)=v(\cdot,T)\) over \(\omega\) then \(u=v\) over \(\Omega\times[0,T]\). If \(g\) is merely locally Lipschitz and the initial data are in \(L^\infty(\Omega)\) then the authors prove an estimate of the form \[ \int_\Omega|(u-v)(x,0)|^2\,dx \leq C\int_\omega|(u-v)(x,T)|^2\,dx. \]
homogeneous Dirichlet boundary conditions
scientific article; zbMATH DE number 5757626