A semilinear parabolic system with a free boundary

Abstract

This paper deals with a semilinear parabolic system with reaction terms $${v^p, u^q}$$ and a free boundary $${x = s(t)}$$ in one space dimension, where $${s(t)}$$ evolves according to the free boundary condition $${s'(t) = -\mu(u_x + \rho v_x)}$$ . The main aim of this paper was to study the existence, uniqueness, regularity and long-time behavior of positive solution (maximal positive solution). Firstly, we prove that this problem has a unique positive solution when $${p, q \geq 1}$$ , and a (unique) maximal positive solution when $${p < 1}$$ or $${q < 1}$$ . Then, we study the regularity of $${(u,v)}$$ and $${s}$$ . At last, we discuss the global existence, finite-time blowup of the unique positive solution (maximal positive solution) and long-time behavior of bounded global solution.