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We investigate the large time behavior of solutions to the following nonlinear parabolic equations { u t − div ( | ∇ u | p ( x ) − 2 ∇ u ) + f ( x , u ) = g in Ω × R + , u = 0 on ∂ Ω × R + , u ( x , 0 ) = u 0 ( x ) in Ω , where u 0 , g ∈ L 1 ( Ω ) . We first provide the existence and uniqueness of an entropy solution for the problem. Then through some delicate analysis, we establish some regularity results on the solution, by which we prove the existence of a global attractor for the solution semigroup. [ABSTRACT FROM AUTHOR]

We consider the obstacle problem related to the following energy with nonstandard growth ∫ Ω | D u | p ( x ) log ⁡ ( e + | D u | ) d x . We investigate the regularity properties of solutions to the obstacle problems along with a suitable assumptions on the variable exponent p ( ⋅ ) and the obstacle. [ABSTRACT FROM AUTHOR]

The aim of this paper is to derive the self-improving property of integrability for the spatial gradient of solutions to degenerate parabolic obstacle problem with irregular obstacles and p ( x , t ) -nonstandard growth. More precisely, we prove that the spatial gradient of the solution is integrable to a larger power than the natural one determined by the structural assumptions on the involved differential operator. [ABSTRACT FROM AUTHOR]

It is shown how delta shock waves which consist of Dirac delta distributions and classical shocks can be used to construct non-monotone solutions of the Buckley–Leverett equation. These solutions are interpreted using a recent variational definition of delta shock waves in which the Rankine–Hugoniot deficit is explicitly accounted for [6] . The delta shock waves are also limits of approximate solutions constructed using a recent extension of the weak asymptotic method to complex-valued approximations [15] . Finally, it is shown how these non-standard shocks can be fitted together to construct similarity and traveling-wave solutions which are non-monotone, but still admissible in the sense that characteristics either enter or are parallel to the shock trajectories. [ABSTRACT FROM AUTHOR]

Abstract: We consider the existence and non-existence of solutions for nonlinear elliptic equations whose model is where Ω is a smooth bounded domain in , μ is a bounded Radon measure. Our results are twofold: when μ is absolutely continuous with respect to the -capacity, the problem admits a unique solution; when μ is concentrated on a set of zero -capacity ( in ), then the problem does not admit a solution if is large enough. [Copyright &y& Elsevier]

Abstract: In this paper, we consider uniform asymptotics for the finite-time ruin probabilities of two kinds of nonstandard bidimensional renewal risk models with constant interest forces and diffusion generated by Brownian motions. In one of the models, two classes of claims have different arrival times, while in the another model, two classes of claims share the same arrival times. In both models, two classes of claim sizes are both upper tail asymptotically independent and their distributions belong to the intersection of the long-tailed distribution class and the dominatedly-varying-tailed distribution class, and the inter-arrival times follow a widely lower orthant dependence structure. In each model, we obtain three kinds of uniform asymptotics for the finite-time ruin probabilities, respectively. [Copyright &y& Elsevier]

Abstract: This paper proposes a nonstandard finite difference (NSFD) method for a class of delay differential equations. We prove that, for any step size (), this numerical method could intrinsically preserve the qualitative behavior of the dynamical system, including the local stability of equilibrium, the existence and the direction of Hopf bifurcation and the stability of bifurcating periodic solution. Finally, to illustrate the analytic results, the NSFD method is used for a model of the survival of red blood cells in animals. [Copyright &y& Elsevier]

Abstract: We characterize Schwartz distributions having a value at a single point in the sense introduced by means of nonstandard analysis by A. Robinson. They appear to be continuous functions in a neighborhood of the point. This characterization improves a result by P. Loeb which assumes the everywhere existence of point values. [Copyright &y& Elsevier]

Abstract: We show that principles from nonstandard analysis hold to some extent for nonlinear generalized functions. The generalized functions under consideration are constructed as families of functions modulo a free filter, as it is usually done in applied analysis. In contrast with models of nonstandard analysis, we do not require the filter to be an ultrafilter. The principles are intended to be used as a tool for proving theorems, which we illustrate by means of an automatic continuity result that was not suspected by experts in the field. [Copyright &y& Elsevier]

Abstract: We prove -regularity for local minimizers of functionals where φ is a Young function. In order to generalize the results of Fuchs (2011) and Diening et al. (2009) , we assume for the function φ only the -condition and for a positive constant (and of course a Hölder-condition for the second derivatives). [Copyright &y& Elsevier]

Abstract: We derive a discretized SIRS epidemic model with time delay by applying a nonstandard finite difference scheme. Sufficient conditions for the global dynamics of the solution are obtained by improvements in discretization and applying proofs for continuous epidemic models. These conditions for our discretized model are the same as for the original continuous model. [Copyright &y& Elsevier]

Abstract: In the present paper we consider a prismatic cylinder occupied by an anisotropic and homogeneous compressible linear thermoelastic material within the framework of the linear theory of thermoelasticity without energy dissipation. The cylinder is subject to zero body force and heat supply and zero lateral specific boundary conditions and the motion is induced by a time-dependent displacement and thermal displacement specified pointwise over the base. Further, the motion is constrained such that the displacement, thermal displacement, velocity and temperature variation at points in the cylinder and at a prescribed time are in given proportions to, but not identical with, their respective initial values. It is shown that certain integrals of the solution spatially evolve with respect to the axial variable. Conditions are derived that show the integrals exhibit alternative behavior and in particular for the semi-infinite cylinder that there is either at least exponential growth or at most exponential decay, provided the elasticity tensor is positive definite or strongly elliptic. [Copyright &y& Elsevier]

Abstract: Under nonstandard growth conditions, following Moser''s iteration technique, we prove boundedness of solutions for fourth order nonlinear elliptic equation in divergence form. [Copyright &y& Elsevier]

We consider problems of the form d2u/dt2+Au=F,αu(0)+u(T)=g,βdu/dt(0)+du/dt+Au=F, αu(0)+u(T)=g, βdu/dt(0)+du/dt(0)+du/dt(T)=h, for t∈(0,T), where A is a densely defined, linear, time independent, positive definite symmetric operator and α and β are constants. Although most of our results would hold for more general operators A, we restrict attention to the case in which A is a differential operator and determine ranges of values of α and β for which it is possible to obtain energy bounds, uniqueness results, and, in a special case, pointwise bounds. Some extensions which include a damping term or a term which arises in a generalization of the Kirchhoff string model are also discussed. [Copyright &y& Elsevier]