### A decomposition theorem for solutions of parabolic equations.

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We prove a generalized maximum principle for subsolutions of boundary value problems, with mixed type unilateral conditions, associated to a degenerate parabolic second-order operator in divergence form.

We propose a multiscale model reduction method for partial differential equations. The main purpose of this method is to derive an effective equation for multiscale problems without scale separation. An essential ingredient of our method is to decompose the harmonic coordinates into a smooth part and a highly oscillatory part so that the smooth part is invertible and the highly oscillatory part is small. Such a decomposition plays a key role in our construction of the effective equation. We show...

We present a simplified approach to the analytical approximation of the transition density related to a general local volatility model. The methodology is sufficiently flexible to be extended to time-dependent coefficients, multi-dimensional stochastic volatility models, degenerate parabolic PDEs related to Asian options and also to include jumps.

In this paper we consider an approximate controllability problem for linear parabolic equations with rapidly oscillating coefficients in a periodically perforated domain. The holes are $\epsilon $-periodic and of size $\epsilon $. We show that, as $\epsilon \to 0$, the approximate control and the corresponding solution converge respectively to the approximate control and to the solution of the homogenized problem. In the limit problem, the approximation of the final state is alterated by a constant which depends on the proportion...

In this paper we consider an approximate controllability problem for linear parabolic equations with rapidly oscillating coefficients in a periodically perforated domain. The holes are ε-periodic and of size ε. We show that, as ε → 0, the approximate control and the corresponding solution converge respectively to the approximate control and to the solution of the homogenized problem. In the limit problem, the approximation of the final state is alterated by a constant which depends on the proportion...

This paper deals with the evolution Fokker-Planck-Smoluchowski configurational probability diffusion equation for the FENE dumbbell model in dilute polymer solutions. We prove the exponential convergence in time of the solution of this equation to a corresponding steady-state solution, for arbitrary velocity gradients.

We study the asymptotic behaviour of a sequence of strongly degenerate parabolic equations ${\partial}_{t}\left({r}_{h}u\right)-\mathrm{div}({a}_{h}\xb7Du)$ with ${r}_{h}(x,t)\ge 0$, ${r}_{h}\in {L}^{\infty}(\Omega \times (0,T))$. The main problem is the lack of compactness, by-passed via a regularity result. As particular cases, we obtain G-convergence for elliptic operators $({r}_{h}\equiv 0)$, G-convergence for parabolic operators $({r}_{h}\equiv 1)$, singular perturbations of an elliptic operator $({a}_{h}\equiv a$ and ${r}_{h}\to r$, possibly $r\equiv 0)$.