Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the behaviour of stochastic heat equations on bounded domains
138
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Consider the following equation $$partial_t u_t(x)=frac{1}{2}partial _{xx}u_t(x)+lambda sigma(u_t(x))dot{W}(t,,x)$$ on an interval. Under Dirichlet boundary condition, we show that in the long run, the second moment of the solution grows exponentially fast if $lambda$ is large enough. But if $lambda$ is small, then the second moment eventually decays exponentially. If we replace the Dirichlet boundary condition by the Neumann one, then the second moment grows exponentially fast no matter what $lambda$ is. We also provide various extensions.
rate research
Read More
In this paper, we obtain upper and lower bounds for the moments of the solution to a class of fractional stochastic heat equations on the ball driven by a Gaussian noise which is white in time, and with a spatial correlation in space of Riesz kernel type. We also consider the space-time white noise case on an interval.
This is the continuation of our previous work [5], where we introduced and studied some nonlinear integral equations on bounded domains that are related to the sharp Hardy-Littlewood-Sobolev inequality. In this paper, we introduce some nonlinear integral equations on bounded domains that are related to the sharp reversed Hardy-Littlewood-Sobolev inequality. These are integral equations with nonlinear term involving negative exponents. Existence results as well as nonexistence results are obtained.
Consider the integral equation begin{equation*} f^{q-1}(x)=int_Omegafrac{f(y)}{|x-y|^{n-alpha}}dy, f(x)>0,quad xin overline Omega, end{equation*} where $Omegasubset mathbb{R}^n$ is a smooth bounded domain. For $1<alpha<n$, the existence of energy maximizing positive solution in subcritical case $2<q<frac{2n}{n+alpha}$, and nonexistence of energy maximizing positive solution in critical case $q=frac{2n}{n+alpha}$ are proved in cite{DZ2017}. For $alpha>n$, the existence of energy minimizing positive solution in subcritical case $0<q<frac{2n}{n+alpha}$, and nonexistence of energy minimizing positive solution in critical case $q=frac{2n}{n+alpha}$ are also proved in cite{DGZ2017}. Based on these, in this paper, the blowup behaviour of energy maximizing positive solution as $qto (frac{2n}{n+alpha})^+ $ (in the case of $1<alpha<n$), and the blowup behaviour of energy minimizing positive solution as $qto (frac{2n}{n+alpha})^-$ (in the case of $alpha>n$) are analyzed. We see that for $1<alpha<n$ the blowup behaviour obtained is quite similar to that of the elliptic equation involving subcritical Sobolev exponent. But for $alpha>n$, different phenomena appears.
We consider the generic divergence form second order parabolic equation with coefficients that are regular in the spatial variables and just measurable in time. We show that the spatial derivatives of its fundamental solution admit upper bounds that agree with the Aronson type estimate and only depend on the ellipticity constants of the equation and the L $infty$ norm of the spatial derivatives of its coefficients. We also study the corresponding stochastic partial differential equations and prove that under natural assumptions on the noise the equation admits a mild solution, given by anticipating stochastic integration.
We consider stochastic heat equations with fractional Laplacian on $mathbb{R}^d$. Here, the driving noise is generalized Gaussian which is white in time but spatially homogenous and the spatial covariance is given by the Riesz kernels. We study the large-scale structure of the tall peaks for (i) the linear stochastic heat equation and (ii) the parabolic Anderson model. We obtain the largest order of the tall peaks and compute the macroscopic Hausdorff dimensions of the tall peaks for both (i) and (ii). These results imply that both (i) and (ii) exhibit multi-fractal behavior in a macroscopic scale even though (i) is not intermittent and (ii) is intermittent. This is an extension of a recent result by Khoshnevisan et al to a wider class of stochastic heat equations.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
