Sufficient conditions for destabilizing effects of certain unilateral boundary conditions and for the existence of bifurcation points for spatial patterns to reaction-diffusion systems of the activator-inhibitor type are proved. The conditions are related with the mollification method employed to overcome difficulties connected with empty interiors of appropriate convex cones.

The main object of this paper is to study the regularity with respect to the parameter h of solutions of the problem $du/dt+A_h\left(t\right)u\left(t\right)=f_h\left(t\right)$, $u\left(0\right)=u_h^0$. The continuity of u with respect to both h and t has been considered in [6].

We consider control problems governed by semilinear parabolic equations with pointwise state constraints and controls in an Lp-space (p < ∞). We construct a correct relaxed problem, prove some relaxation results, and derive necessary optimality conditions.

We consider control problems governed by semilinear parabolic equations with pointwise state constraints and controls in an $L^p$-space ($p\&lt;\infty$). We construct a correct relaxed problem, prove some relaxation results, and derive necessary optimality conditions.