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In our work we study non-variational, nonlinear singularly perturbed elliptic models enjoying a double degeneracy character with prescribed boundary value in a domain. In such a scenario, we establish the existence of solutions. We also prove that solutions are locally (uniformly) Lipschitz continuous, and they grow in a linear fashion. Moreover, solutions and their free boundaries possess a sort of measure-theoretic and weak geometric properties. Moreover, for a restricted class of non-linearities, we prove the finiteness of the (N-1)-dimensional Hausdorff measure of level sets. We also address a complete analysis concerning the asymptotic limit as the singular parameter, which is related to one-phase solutions of inhomogeneous nonlinear free boundary problems in flame propagation and combustion theory.
Given a symmetric Riemannian manifold (M, g), we show some results of genericity for non degenerate sign changing solutions of singularly perturbed nonlinear elliptic problems with respect to the parameters: the positive number {epsilon} and the symm
An averaging method is applied to derive effective approximation to the following singularly perturbed nonlinear stochastic damped wave equation u u_{tt}+u_t=D u+f(u)+ u^alphadot{W} on an open bounded domain $DsubsetR^n$,, $1leq nleq 3$,. Here $ u>0
We explore the relation between fast waves, damping and imposed noise for different scalings by considering the singularly perturbed stochastic nonlinear wave equations u u_{tt}+u_t=D u+f(u)+ u^alphadot{W} on a bounded spatial domain. An asymptoti
We consider singularly perturbed convection-diffusion equations on one-dimensional networks (metric graphs) as well as the transport problems arising in the vanishing diffusion limit. Suitable coupling condition at inner vertices are derived that gua
We consider smooth systems limiting as $epsilon to 0$ to piecewise-smooth (PWS) systems with a boundary-focus (BF) bifurcation. After deriving a suitable local normal form, we study the dynamics for the smooth system with $0 < epsilon ll 1$ using a c