Global existence for a semilinear differential system
Let \(\Omega\) be a bounded domain with smooth boundary \(\partial \Omega\) in the \((x_1, x_2)\)-plane. Consider the problem of the system of elliptic-parabolic partial differential equations arising in a two-dimensional groundwater flow problem: \[ - \Delta v= {\partial u\over \partial x_1}\quad \text{in } (0, \infty)\times \Omega,\quad v= 0\quad \text{on } (0, \infty)\times \partial \Omega,\tag{E} \] \[ {\partial u\over \partial t}- \Delta u+ \text{grad } u\cdot \text{curl } v= 0\text{ in } (0, \infty)\times \Omega,\;{\partial u\over \partial \vec \nu}= 0\text{ on } (0, \infty)\times \partial \Omega,\;u(0, \cdot)= u_0(\cdot),\tag{P} \] where \(\vec \nu\) denotes the outer normal unit vector on \(\partial \Omega\) and \(u_0\) is a given function on \(\Omega\). By setting the problem (E), (P) in the abstract form the author establishes \(L^p- L^q\) a priori estimates for solutions of the linearized problems. Then the global existence and uniqueness of solutions for the problem (E), (P) are proved.
scientific article; zbMATH DE number 850813