Self-similar solutions for a quenching problem with spatially dependent nonlinearity

This paper is devoted to the study of the self-similar solutions for a semilinear parabolic equation with spatially dependent nonlinearity which arises in the model of micro-electro mechanical system. We first provide a result on the non-existence of slow orbit for a certain range of parameters. Next, we prove the existence of backward solutions with the desired polynomial growth condition at infinity to the associated equation by a fixed point argument. Then we give a detailed analysis of the behavior of global solutions at the origin. Finally, as an application of the above results, we prove a uniqueness theorem.