Existence of regular solutions for semilinear parabolic integro- differential equations
The author studies existence and regularity problems for the semilinear parabolic equation \[ (1)\quad u_ t-\Delta_ xu+a*g(u)=f,\quad (x,t)\in \Omega \times [0,T] \] (2) \(u\equiv 0\), (x,t)\(\in \partial \Omega \times [0,T]\). Here equation (1) can be considered as an extension of the equation \[ (3)\quad u_ t-\Delta_ xu+g(u)=f,\quad (x,t)\in \Omega \times [0,T] \] with boundary condition (2). The crucial growth restriction on g that the author needs to establish the existence of a solution of (1), coincides with the weakest known properties of g that guarantee the existence of a solution of (3). Furthermore, the equation (4) \(u_ t-\Delta_ xu=f\), \((x,t)\in \Omega [0,T]\) with boundary condition (5) \(\partial_ vu+a*g(u)=h\), \((x,t)\in \partial \Omega \times [0,T]\) is studied. The boundary condition (4) is an extension of the condition \(\partial_ vu+g(u)=h\), \((x,t)\in \partial \Omega \times [0,T].\) For the problem (4) and (5) the regularizing effect (in time) of the integral operator is also used. Both scalar and vector equations are considered.
scientific article; zbMATH DE number 3979749
scientific article