No Arabic abstract
In this paper we obtain the precise description of the asymptotic behavior of the solution $u$ of $$ partial_t u+(-Delta)^{frac{theta}{2}}u=0quadmbox{in}quad{bf R}^Ntimes(0,infty), qquad u(x,0)=varphi(x)quadmbox{in}quad{bf R}^N, $$ where $0<theta<2$ and $varphiin L_K:=L^1({bf R}^N,,(1+|x|)^K,dx)$ with $Kge 0$. Furthermore, we develop the arguments in [15] and [18] and establish a method to obtain the asymptotic expansions of the solutions to a nonlinear fractional diffusion equation $$ partial_t u+(-Delta)^{frac{theta}{2}}u=|u|^{p-1}uquadmbox{in}quad{bf R}^Ntimes(0,infty), $$ where $0<theta<2$ and $p>1+theta/N$.
We examine the short and long-time behaviors of time-fractional diffusion equations with variable space-dependent order. More precisely, we describe the time-evolution of the solution to these equations as the time parameter goes either to zero or to infinity.
The large time behavior of zero mass solutions to the Cauchy problem for a convection-diffusion equation. We provide conditions on the size and shape of the initial datum such that the large time asymptotics of solutions is given either by the derivative of the Guass-Weierstrass kernel or by a self-similar solution or by a hyperbolic N-wave
We obtain asymptotic mean value formulas for solutions of second-order elliptic equations. Our approach is very flexible and allows us to consider several families of operators obtained as an infimum, a supremum, or a combination of both infimum and supremum, of linear operators. We study both when the set of coefficients is bounded and unbounded (each case requires different techniques). The families of equations that we consider include well-known operators such as Pucci, Issacs, and $k-$Hessian operators.
This paper deals with the investigation of the computational solutions of an unified fractional reaction-diffusion equation, which is obtained from the standard diffusion equation by replacing the time derivative of first order by the generalized fractional time-derivative defined by Hilfer (2000), the space derivative of second order by the Riesz-Feller fractional derivative and adding the function phi(x,t) which is a nonlinear function overning reaction. The solution is derived by the application of the Laplace and Fourier transforms in a compact and closed form in terms of the H-function. The main result obtained in this paper provides an elegant extension of the fundamental solution for the space-time fractional diffusion equation obtained earlier by Mainardi et al. (2001, 2005) and a result very recently given by Tomovski et al. (2011). Computational representation of the fundamental solution is also obtained explicitly. Fractional order moments of the distribution are deduced. At the end, mild extensions of the derived results associated with a finite number of Riesz-Feller space fractional derivatives are also discussed.
We propose two asymptotic expansions of the two interrelated integral-type averages, in the context of the fractional $infty$-Laplacian $Delta_infty^s$ for $sin (frac{1}{2},1)$. This operator has been introduced and first studied in [Bjorland-Caffarelli-Figalli, 2012]. Our expansions are parametrised by the radius of the removed singularity $epsilon$, and allow for the identification of $Delta_infty^sphi(x)$ as the $epsilon^{2s}$-order coefficient of the deviation of the $epsilon$-average from the value $phi(x)$, in the limit $epsilonto 0+$. The averages are well posed for functions $phi$ that are only Borel regular and bounded.