a regularity theorem for a nonconvex scalar conservation law abstract in this paper we study the regularity properties of solutions of a single conservation law we prove that if the flux function f is smooth and totally nonlinear in the sense that f vanishes at isolated points only and if the initial data u0 are bounded and measurable then f u t is in the class of functions of locally bounded variation for all t 0

bounds for blow up time in nonlinear parabolic problems abstract a first order differential inequality technique is used on suitably defined auxiliary functions to determine lower bounds for blow up time in initial boundary value problems for parabolic equations of the form u t div u grad u f u if blow up occurs in addition conditions which ensure that blow up occurs or does not occur are presented

weak solutions of the landau lifshitz bloch equation abstract the landau lifshitz bloch llb equation is a formulation of dynamic micromagnetics valid at all temperatures treating both the transverse and longitudinal relaxation components important for high temperature applications we study llb equation in case the temperature raised higher than the curie temperature the existence of weak solution is showed and its regularity properties are also discussed in this way we lay foundations for the rigorous theory of llb equation that is currently not available

computational studies for the stochastic landau lifshitz gilbert equation the stochastic landau lifshitz gilbert equation describes the thermally induced dynamics of magnetic moments in ferromagnetic materials solutions of this highly nonlinear stochastic pde are unit vector fields and satisfy an energy estimate these are crucial properties to construct a convergent discretization in space and time we propose a convergent finite element approximation of the problem based on the midpoint rule the numerical scheme preserves the underlying properties of the continuous problem further we construct a robust and efficient newton multigrid solver for the solution of the nonlinear systems associated with the discretized problems at each time level computational studies show the optimal convergence behavior of the scheme in the case of smooth solutions long time dynamics for finite ensembles of spins evidence the ergodicity of an invariant measure of the continuum model numerical experiments in two dimensions demonstrate pathwise finite time blow up behavior of the solution w

splitting integrators for the stochastic landau lifshitz equation in this article we construct splitting integrators for a finite dimensional version of the stochastic landau lifshitz equation under the influence of global and local energy terms the methods preserve the length of the magnetization spins exactly and reproduce the energy evolution of the equation depending on the structure of the hamiltonian the methods either are explicit or contain nonlinear subsystems of small dimension which leads to a smaller computational cost compared with standard implicit integrators numerical simulations are provided for evidencing convergence order and long time behavior of the constructed methods

bounds for blow up time of a reaction diffusion equation with weighted gradient nonlinearity abstract in this paper we focus on the bounds for blow up time of null dirichlet initial boundary value problem for a reaction diffusion equation with weighted gradient nonlinearity by virtue of the method of super sub solution and the technique of modified differential inequality we establish sufficient conditions to guarantee that the solution blows up at finite time under appropriate measure sense meanwhile upper and lower bounds for the blow up time are found in higher dimensional spaces and some examples for application are presented

martingale weak solutions of the stochastic landau lifshitz bloch equation abstract the present paper studies the stochastic landau lifshitz bloch equation which is recommended as the only valid model at temperature around the curie temperature and is especially important for the simulation of heat assisted magnetic recording we study the stochastic landau lifshitz bloch equation in the case that the temperature is raised higher than the curie temperature the global existence of martingale weak solutions is proved by using a new argument and regularity properties of the weak solutions are discussed

convergence of finite element solutions of stochastic partial integro differential equations driven by white noise numerical approximation of a stochastic partial integro differential equation driven by a space time white noise is studied by truncating a series representation of the noise with finite element method for spatial discretization and convolution quadrature for time discretization sharp order convergence of the numerical solutions is proved up to a logarithmic factor numerical examples are provided to support the theoretical analysis

blow up phenomena for a parabolic problem with a gradient nonlinearity under nonlinear boundary conditions this paper is concerned with the blow up of the solution to a parabolic problem with a gradient nonlinearity under nonlinear boundary conditions lower bound for blow up time is derived when the blow up does occur

blow up of solutions of nonlinear heat equations on considere des solutions de u t u xx f u pour a 0 u a t 0 pour t 0 u x 0 x pour a x a ou f u est une fonction superlineaire continument differentiable

explicit lower bound of blow up time for an attraction repulsion chemotaxis system abstract in this paper we study classical solutions to the zero flux attraction repulsion chemotaxis system u t u u v u w in 0 t 0 v u v in 0 t 0 w u w in 0 t where is a smooth and bounded domain of r 2 t is the blow up time and are positive real numbers from the literature it is known that under a proper interplay between the above parameters and suitable assumptions on the initial data u x 0 u 0 c 0 system has a unique classical solution which becomes unbounded as t t the main result of this investigation is to provide an explicit lower bound for t estimated in terms of u 0 2 d x and attained by means of well established techniques based on ordinary differential inequalities

blow up phenomena for some nonlinear parabolic problems abstract we consider the blow up of solutions of equations of the form u t div u 2 grad u f u by means of a differential inequality technique a lower bound for blow up time is determined if blow up does occur as well as a criterion for blow up and conditions which ensure that blow up cannot occur

self similar subsolutions and blow up for nonlinear parabolic equations abstract for a wide class of nonlinear parabolic equations of the form u t u f u u we give finite time blow up results for large initial data blow up time estimates are also provided these results rely on a new method of comparison with suitable blowing up self similar subsolutions as a consequence we improve several known results on reaction diffusion equations on generalized burgers equations and on the equation of chipot and weissler this method can also apply to degenerate equations of porous medium type and provides a unified treatment for a large class of problems both semilinear and quasilinear

a finite element approximation for the stochastic landau lifshitz gilbert equation abstract the stochastic landau lifshitz gilbert llg equation describes the behaviour of the magnetisation under the influence of the effective field containing random fluctuations we first transform the stochastic llg equation into a partial differential equation with random coefficients without the ito term the resulting equation has time differentiable solutions we then propose a convergent linear scheme for the numerical solution of the reformulated equation as a consequence we show the existence of weak martingale solutions to the stochastic llg equation a salient feature of this scheme is that it does not involve solving a system of nonlinear algebraic equations and that no condition on time and space steps is required when 1 2 1 numerical results are presented to show the applicability of the method

a bound for global solutions of semilinear heat equations we show that global positive solutions of the initial boundary value problem foru t u u p are bounded provided thatp 1 is subcritical our bound depends only on sup norm of the initial data and is useful to classify initial data by the asymptotic behavior of the solutions as time tend to infinity

lower bounds for blow up time in parabolic problems under dirichlet conditions we consider an initial boundary value problem for the semilinear heat equation whose solution may blow up in finite time we use a differential inequality technique to determine a lower bound on blow up time if blow up occurs a second method based on a comparison principle is also presented

wong zakai approximation for the stochastic landau lifshitz gilbert equations abstract in this work we study stochastic landau lifshitz gilbert equations sllges in one dimension with non zero exchange energy only firstly by introducing a suitable transformation we convert the sllges to a highly nonlinear time dependent partial differential equation with random coefficients which is not fully parabolic we then prove that there exists a pathwise unique solution to this equation and that this solution enjoys the maximal regularity property following regular approximation of the brownian motion and using reverse transformation we show existence of strong solution of sllges taking values in a two dimensional unit sphere s 2 in r 3 the construction of the solution and its corresponding convergence results are based on wong zakai approximation

rate estimates of gradient blowup for a heat equation with exponential nonlinearity abstract we consider a one dimensional semilinear parabolic equation u t u x x e u x for which the spatial derivative of solutions becomes unbounded in finite time while the solutions themselves remain bounded we establish estimates of blowup rate upper and lower bounds we prove that in this case the blowup rate does not match the one obtained by the rescaling method

lower bounds for blow up time in a nonlinear parabolic problem abstract for a parabolic problem with a gradient nonlinearity which was introduced by chipot and weissler m chipot f b weissler some blow up results for a nonlinear parabolic problem with a gradient term siam j math anal 20 1989 886 907 see also b kawohl l a peletier observations on blow up and dead cores for nonlinear parabolic equations math z 202 1989 207 217 the question of blow up is investigated specifically if the solution blows up a lower bound for the time of blow up is derived

the cauchy problem for ut δu u q with q a positive real number the nonlinear partial differential equation in the title of the paper arises in the study of the growth of surfaces in that context it is known as the generalized deterministic kpz equation the paper is concerned with the initial value problem for the equation under the assumption that the initial data function is bounded and continuous results on the existence uniqueness and regularity of solutions are obtained

the dynamics of rotating waves in scalar reaction diffusion equations on etudie l attracteur compact maximal pour l equation u t u xx f u u x avec des conditions aux limites periodiques on montre qu un ensemble limite contient une onde de rotation

blow up time estimate for a degenerate diffusion equation with gradient absorption this paper deals with a degenerate nonlinear diffusion equation with gradient absorption we at first determine finite time blow up of solutions both in the l norm and an integral measure and then estimate a lower bound of the blow up time by using the differential inequality technique it is mentioned that the blowing up of solutions to nonlinear pdes is usually defined in the l norms while the lower bounds of blow up time are all determined via some measures in the form of energy integrals so in general to estimate the lower bounds of blow up time it has to be assumed that the solutions do blow up in finite time with the involved integral measure before establishing their lower bounds of blow up time such assumptions are unnecessary in this paper