We develop a solution theory in Hölder spaces for a quasi-linear stochastic PDE driven by an additive noise. The key ingredients are two deterministic PDE lemmas which establish a priori Hölder bounds for a parabolic equation in divergenceform with irregular right-hand-side term. We apply these bounds to the case of a right-hand- side noise term which is white in time and trace class in space, to obtain stretched exponential bounds for the Hölder semi-norms of the solution.
Abstract The Zeldovich-Barenblatt type solution of the Cauchy problem for a cross-diffusion parabolic system not in divergenceform with a source and a variable density is obtained. Based on comparison method the property of finite speed perturbation of distribution is considered. An asymptotic behavior of self-similar solutions, both for slow and fast diffusion cases, is established. It is obtained the system of the nonlinear algebraic equations with the coefficients of the main terms of the asymptotical solution.
on Riemannian manifolds, we refer to [–] and the references therein. As brieﬂy men- tioned above, it is a natural problem how to get the universal inequalities of the eigenvalues of elliptic operator in weighted divergenceform. Actually, in this paper, we ﬁrst consider the eigenvalue problem as follows:
Programs for the numerical solution of nonlinear systems not in divergenceform developed in MATLAB. Programs are compact. By the user is entered the necessary numerical parameters. At the end of the file automatically displays the calculated results in the form of matrices and graphic. In the same place by running animation can trace the evolution of the process in time.
precisely the subject of this paper. Once again, the key point is the construction of the previously mentioned family of subsolutions. Unlike the case of nondivergence or fully nonlinear operators, in the case of divergenceform operators, the construction turns out to be rather delicate due to the fact that in this case not only the quadratic part of a function controls in average the action of the operator but also the linear part has an equivalent influence. Here we require Lipschitz continuous coeﬃcients.
method for diﬀerential equations of divergenceform to prove the existence of weak so- lutions for (.) with Dirichlet boundary or Neumann boundary condition in separable Musielak-Orlicz-Sobolev spaces. We give the enclosure of weak solutions between sub- and supersolutions by using a sub-supersolution method. Our method does not require any monotonicity or coercivity of a . We point out that the coercive condition (.) of
We study the Dirichlet problem for linear elliptic second order partial diﬀerential equations with discontinuous coeﬃcients in divergenceform in unbounded domains. We establish an existence and uniqueness result and we prove an a priori bound in L p , p > 2.
are studied. In this work used: method of nonlinear splitting, known previously for non-linear parabolic equations and systems of equations in divergenceform, asymptotic theory and asymptotic methods based on diﬀerent transformations. Constructed asymptotic representation of self-similar solutions of nonlinear parabolic systems of equations not in divergenceform, depending on the value in the system of the numerical parameters necessary and suﬃcient signs of their existence. The main purpose of this paper is to find conditions for the existence and non-existence results for global solutions of parabolic equations not in divergenceform on the basis of the self-similar analysis. Keywords nonlinear parabolic systems of equations, not in divergenceform, global solutions, self-similar solutions, asymptotic representation of solution
Abstract. We establish quantitative results on the periodic approximation of the corrector equation for the stochastic homogenization of linear elliptic equations in divergenceform, when the diffusion coefficients satisfy a spectral gap estimate in probability, and for d > 2. This work is based on , which is a complete continuum version of [6, 7] (with in addition optimal results for d = 2). The main difference with respect to the first part of  is that we avoid here the use of Green’s functions and more directly rely on the De Giorgi-Nash-Moser theory.
a trivial Carnot group. In the Euclidean case, we ﬁrst recall that Zhang  studied the global existence for a parabolic problem in divergenceform analogous to (.) when the potential V is in parabolic Kato class at inﬁnity P ∞ , the asymptotic behavior of solutions for the problem was studied by Zhang and Zhao . Riahi  extended the results in 
Divergence measures have been demonstrated very useful in a variety of disciplines such as economics and political science (Gokhale and Kullback, 1978), ), signal processing (Kadota and (Bassat, 1978; Chen, 1973 and Jones and Byrne, 1990), color image (Taskar et al., 2006), cost- sensitive , magnetic resonance image analysis (Vemuri et al., 2010) etc. Also we can use divergences in fuzzy mathematics as fuzzy directed divergences and fuzzy entropies (Bajaj and Hooda, 2010;
In whole paper, in the section 2, we have introduced information inequalities. New non-symmetric information divergence measure has derived in section 3. Bounds of new information divergence measure in terms of Kullback-Leibler divergence measure have studied in section 4. In section 5, give the numerical bounds of new non-symmetric information divergence in terms of Kullback-Leibler divergence measure.