Behaviour of interfaces in a diffusion-absorption equation with critical exponents (Q1590208)

The authors investigate the nonlinear diffusion equation \(u_t= (u^m)_{xx}- u^p\) for \(x\in\mathbb{R}\), \(t> 0\), with continuous compactly supported initial data \(u(x,0)= u_0(x)\geq 0\) in the critical case \(m+ p=2\), \(m> 1\), \(p< 1\). Physically, the equation describes thermal propagation in diffusive-absorbing porous media. The main aim is to study the regularity of the ``interface'', i.e. the curve separating the regions \(\{u>0\}\) and \(\{u= 0\}\) in the plane \((x,t)\). Using the maximum principle and matching technique, the authors establish the analyticity of the interface and demonstrate the occurrence of turning as well as inflection points for the interface curves. The results are compared with the purely diffusive case \(u_t= (u^m)_{xx}\) and some qualitative differences are found.