Blow-Up of the Solution for some Higher Order Hyperbolic and Neutral Evolution Systems

Time analyticity for solutions of nonlinear abstract parabolic evolution equations

In this paper we study spaces of functions on domains with a conical singularity, which arise naturally if we study hyperbolic and parabolic evolution equations on such domains. We state conditions under which these spaces are Banach algebras, a result which is of independent interest. As applications, we remark that this implies local existence of solutions of Cauchy problems for semilinear, hyperbolic and parabolic partial differential equations with zero Dirichlet boundary conditions and a power-nonlinearity.

In this paper we are interested in a nonlinear parabolic evolution equation occurring in rheology. We give a probabilistic interpretation to this equation by associating a nonlinear martingale problem with it. We prove the existence of a unique solution, P, to this martingale problem. For any t, the time marginal of P at time t admits a density ρ(t,x) with respect to the Lebesgue measure, where the function ρ is the unique weak solution to the evolution equation in a well-chosen energy space. Next we introduce a simulable system of n interacting particles and prove that the empirical measure of this system converges to P as n tends to ∞. This propagation-of-chaos result ensures that the solution to the equation of interest can be approximated using a Monte Carlo method. Finally, we illustrate the convergence in some numerical experiments.

Large time behaviour of Hele–Shaw flow with surface tension and with injection or suction in one point is discussed. We consider domains that are initially small perturbations of balls. Radially, symmetric solutions are stationary after a suitable time-dependent rescaling. The evolution of perturbations can be described by a non-local non-linear parabolic evolution equation. Global existence results and decay properties are derived using energy estimates in Sobolev spaces.

We investigate a general control problem for a class of nonlinear parabolic evolution equations. Applications are related to solid-solid and solid-liquid phase transitions. We prove compactness of the solution operators, existence of optimal controls and show convergence of the finite-dimensional approximate control problem to the original one.

We consider nonlinear parabolic evolution equations of the form $\partial_{t}u=F(t,x,Du,D^{2}u) $, subject to noise of the form $H(x,Du) \circ dB$ where $H$ is linear in $Du$ and $\circ dB$ denotes the Stratonovich differential of a multidimensional Brownian motion. Motivated by the essentially pathwise results of [Lions, P.-L. and Souganidis, P.E.; Fully nonlinear stochastic partial differential equations. C. R. Acad. Sci. Paris S\'{e}r. I Math. 326 (1998), no. 9] we propose the use of rough path analysis [Lyons, T. J.; Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 (1998), no. 2, 215--310] in this context. Although the core arguments are entirely deterministic, a continuity theorem allows for various probabilistic applications (limit theorems, support, large deviations, ...).

In the article, using the modified Levy method, a Green's function for a class of ultraparabolic equations of high order with an arbitrary number of parabolic degeneration groups is constructed. The modified Levy method is developed for high-order Kolmogorov equations with coefficients depending on all variables, while the frozen point, which is a parametrix, is chosen so that an exponential estimate of the fundamental solution and its derivatives is conveniently used.

Stationary self-propagating fronts in potential flow

Averaging principle and hyperbolic evolution equations

Stochastic differential delay equations and their asymptotic behavior have received much attention in recent years (see [1, 2, 5, 7–10, 12, 13] and the references therein), since these retarded problems often appear in physics, biology, engineering, etc. The delays can enter into the formulations in very different ways, e.g., as a constant or variable delay or as a distributed one; some of them can even appear in the model at the same time. However, it is possible to consider all of them under a unified formulation by using appropriate differential functional equations. In this sense, we will carry out our analysis in a functional framework that will cover a wide variety of situations containing finite delays. There exists a vast literature concerning pathwise exponential stability of parabolic stochastic evolution equations (with and without delays). We mention here, among many others, [4, 11–13] and the references therein. In the case without hereditary characteristics, the problem of the asymptotic stability of delay stochastic partial differential equations of second order in time has been considered by Curtain in [6], where one can find sufficient conditions for the exponential stability of the expected energy of the system, as well as for the exponential decay of the sample paths, when the main operator generates a strongly continuous contraction semigroup. In this paper, we shall develop the theory in a variational framework for nonlinear operators in general. We are going to consider the following model:

where C and a are some positive constants, is uniquely solvable. Similar classes were constructed for general linear parabolic systems in [2], for semilinear second-order parabolic equations in [3], and for quasilinear higher-order parabolic equations in [4–7]; moreover, the last papers show the dependence of well-posedness classes for mixed problems in unbounded domains on geometric properties of the domains. Later, it turned out that the unique solvability of the Cauchy problem for some semilinear and nonlinear parabolic equations is independent of the behavior of the solution as |x| → +∞ [8–20]. This effect was essentially obtained for the first time in [8] for the spatially one-dimensional equation ut −∆ ( |u|p−2u ) + c|u|q−2u = 0 (1)

On non-linear evolution equations of higher order – the introduction and application of a novel computational approach

We study the phenomenon of the infinite-time stabilization of classical global solutions of nonlinear reaction–diffusion equations to an unbounded (singular) stationary state and we present a new case of such an asymptotic singularity pattern formation. We concentrate on the most famous parabolic model, namely the semi-linear Frank-Kamenetskii equation from combustion theory, where B is a ball in ℝN. Our goal is to show that a new asymptotic problem arises precisely in dimension 10, not being available in other dimensions (which were studied earlier). For N = 10, we fix the ball B = {|x| < 4} and take bounded initial data u0 below the singular stationary solution Us(x) = ln(16/|x|2), which is unbounded at the origin x = 0.We establish a sharp estimate on the rate of convergence u(x, t) → Us(x) as t → ∞ on compact subsets bounded away from x = 0 and also at the singularity point. We show that u(0, t) = α0t + O(ln t) → ∞, where the positive constant α0 is given by the first eigenvalue of the associated linear differential operator. We present formal asymptotic results showing that a detailed asymptotic analysis depends on a quite involved balance between various linear and nonlinear terms. Moreover, similar critical asymptotic behaviour is shown to exist in various related nonlinear second- and higher-order parabolic equation.

Proper characterization of underwater acoustic transmission loss (TL) relies on proper treatment of the air/water and water/sediment interfaces. This paper will focus on how the air/water interface is treated in parabolic equation (PE) models such as the range dependent acoustic model (RAM) [Collins, M. D., ¿Applications and time-domain solution of higher-order parabolic equations in underwater acoustics,¿ J. Acoust. Soc. Am., 86 (3), 1097-1102, 1989] and the finite element parabolic equation (FEPE) model [Collins, M.D., "A higher-order parabolic equation for wave propagation in an ocean overlying an elastic bottom", J. Acoust. Soc. Am. 86, 1459-1464, 1989]. Two surface loss methods have been implemented in RAM, a surface loss versus angle (LVA) and a conformal mapping (CM) method. Additionally, the CM method has been implemented in FEPE. These two methods are discussed and compared for realistic test cases and a third option, which has accuracy consistent with the CM but speed more aligned with the LVA approach is discussed.

Techniques are developed for projecting the solutions of symmetric hyperbolic evolution systems onto the constraint submanifold (the constraint-satisfying subset of the dynamical field space). These optimal projections map a field configuration to the ``nearest'' configuration in the constraint submanifold, where distances between configurations are measured with the natural metric on the space of dynamical fields. The construction and use of these projections is illustrated for a new representation of the scalar field equation that exhibits both bulk and boundary generated constraint violations. Numerical simulations on a black-hole background show that bulk constraint violations cannot be controlled by constraint-preserving boundary conditions alone, but are effectively controlled by constraint projection. Simulations also show that constraint violations entering through boundaries cannot be controlled by constraint projection alone, but are controlled by constraint-preserving boundary conditions. Numerical solutions to the pathological scalar field system are shown to converge to solutions of a standard representation of the scalar field equation when constraint projection and constraint-preserving boundary conditions are used together.