Large-time behaviour of solutions of the exterior-domain Cauchy-Dirichlet problem for the porous media equation with homogeneous boundary data
The first part of the paper concerns the following radially-symmetric problem: \[ u_t=r^{1-n}(r^{n-1}(u^m)_r)_r\quad (r,t)\in (1,\infty)\times \mathbb{R_+} ,\quad m>1; \] \[ u(1,t)=0\quad \forall\, t \in \mathbb{R_+}; \quad u(r,0)=u_0(r) \quad \forall\, r \in (1,\infty). \] It is show that in time solution \(u(r,t)\) converges to: 1) the self-similar source-type Zel'dovich-Kompaneets-Barenblatt-Pattle solution \(U_s(r,t;a)\) with some constant \(a>0\), if \(n>2\); 2) the self-similar dipole-type Barenblatt-Zel'dovich-King solution \(U_d(r;t;c)\) with some constant \(c>0\), if \(n<2\); 3) \(U_s(r,t;b(\ln{t})^{-\frac{m-1}{2m}})=U_d(r,t;b(\ln{t})^{-\frac{m-1}{2m}})\) with some constant \(b>0\), if \(n=2\). Moreover, the free boundary connected with support of solution converges to that of the corresponding self-similar solution. The technique used is a comparison principle involving a variable that is a weighted integral of the solution. The second part of the paper is devoted to the problem in an arbitrary spatial domain with no condition of symmetry. A special invariance principle and results obtained for the radially-symmetric case are used to determine the large-time behaviour of solutions and their free boundaries.
scientific article; zbMATH DE number 5133243