On semilinear parabolic problems with non-Lipschitz nonlinearities (Q2713035)

Let \(\Omega\) be a bounded domain in \(\mathbb R^N\) or \(\Omega=\mathbb R^N\) and let \(g:[0,\infty)\to[0,\infty)\) be continuous. Consider the equation \(u_t-\Delta u=g(u)\) on \((0,T)\times\Omega\), complemented by the homogeneous Dirichlet boundary condition if \(\Omega\) is bounded and by the initial condition \(u(0,x)=u_0(x)\geq 0\), \(u_0\in L^\infty(\Omega)\). The author proves several comparison principles and existence-uniqueness results for positive mild solutions of this problem under various additional assumptions on \(g\). The comparison principles require \(g(u)-g(v)\leq C_1(M)(u-v)/v\) (or \(g(u)-g(v)\geq C_2(M)(u-v)\)) for all \(0<v\leq u\leq M\). The existence-uniqueness results require, in addition, \(g\) to be concave on \((0,a)\) for some \(a>0\), \(g'(0)=\infty\), and \(\int_0^b 1/g(s) ds<\infty\) for some \(b>0\). If \(\int_0^b 1/g(s) ds=\infty\) for all \(b>0\) then the existence of a unique positive (maximal) solution is true only for \(u_0\not\equiv 0\) (and \(u\equiv 0\) is the only solution if \(u_0\equiv 0\)). A typical example is represented by the function \(g(u)=\lambda(u^q+u^p)\), where \(0<q<1<p\) and \(\lambda>0\).