We consider the second order Cauchy problem εu ε + u ε + m A 1/2 u ε 2 Au ε = 0, u ε (0) = u 0 , u ε (0) = u 1 , and the first order limit problemand m : [0, +∞) → [0, +∞) is a function of class C 1 .We prove decay estimates (as t → +∞) for solutions of the first order problem, and we show that analogous estimates hold true for solutions of the second order problem provided that ε is small enough. We also show that our decay rates are optimal in many cases.The abstract results apply to parabolic and hyperbolic partial differential equations with nonlocal nonlinearities of Kirchhoff type.

We consider an abstract second order linear equation with a strong dissipation, namely a friction term which depends on a power of the "elastic" operator.In the homogeneous case, we investigate the phase spaces in which the initial value problem gives rise to a semigroup, and the further regularity of solutions. In the nonhomogeneous case, we study how the regularity of solutions depends on the regularity of forcing terms, and we characterize the spaces where a bounded forcing term yields a bounded solution.What we discover is a variety of different regimes, with completely different behaviors, depending on the exponent in the friction term.We also provide counterexamples in order to show the optimality of our results.Mathematics Subject Classification 2010 (MSC2010): 35L10, 35L15, 35L20.

We consider a class of semi-linear dissipative hyperbolic equations in which the operator associated to the linear part has a nontrivial kernel.Under appropriate assumptions on the nonlinear term, we prove that all solutions decay to 0, as t → +∞, at least as fast as a suitable negative power of t. Moreover, we prove that this decay rate is optimal in the sense that there exists a nonempty open set of initial data for which the corresponding solutions decay exactly as that negative power of t.Our results are stated and proved in an abstract Hilbert space setting, and then applied to partial differential equations.

We consider the Cauchy problem for the Perona-Malik equationin a bounded open set ⊆ R n , with Neumann boundary conditions.If n = 1, we prove some a priori estimates on u and u x . Then we consider the semi-discrete scheme obtained by replacing the space derivatives by finite differences. Extending the previous estimates to the discrete setting we prove a compactness result for this scheme and we characterize the possible limits in some cases. Finally, for n > 1 we give examples to show that the corresponding estimates on ∇u are in general false.

Abstract. We consider the Cauchy problem for the one-dimensional PeronaMalik equation, with homogeneous Neumann boundary conditions.We prove that the set of initial data for which this equation has a localin-time classical solution u :