Hamiltonian structure of ``Thirring \(\times\) Liouville'' model. Singular solutions
In two-dimensional space-time, interacting scalar and spinor fields possessing definite singularities in the (t,x) plane are cn (- \(\infty,0]\), \(g(-1)=0\), g(u)\(\leq 0\) on (-\(\infty,-1]\) 0\(\leq \ell \leq 1\), \(\phi \in H^ 1_ 0(0,1)\), \(\phi (\ell)=0\), (\(\ell - x)\phi (x)\geq 0,0\leq x\leq 1.\) This problem of the Stefan type with sources was studied by the authors in previous papers, under non-homogeneous Dirichlet data, where the free boundary s(t) never touches \(x=0,1\). In the present case it is possible that one phase disappear in finite time. The authors study global existence, uniqueness, regularity and asymptotic properties of solutions. Attention is given to the possibility that \(s(T^*)\) be either 0 or 1 for certain \(T^*:\) the authors show that in that case s(t) will be identically 0 or 1 for \(t\geq T^*\), and therefore the problem becomes an initial-boundary value problem for \(t\geq T^*\). Comparison theorems are given, and results on the dependence of solutions on the initial data \(\phi\),\(\ell\) are proven. These results are used to study the \(\omega\)-limit set corresponding to the solution orbit of the problem: every element in this set satisfies the associated stationary problem. Stability and bifurcation are also studied.
essentially nonlinear evolution equations
dependence of solutions on the initial data
scientific article; zbMATH DE number 4059984