We consider the stochastic heat equation $\partial_t u(t,x) = \Delta u(t,x) + \dot{W}_\alpha(t,x)$ on a bounded Lipschitz domain with zero Dirichlet boundary condition and zero initial condition, where $\dot{W}_\alpha$ is a Gaussian noise that is white in time and whose spatial covariance is the kernel of $(-\Delta)^{-\alpha}$ with $\alpha>0$. We prove that a unique pointwise defined mild solution exists if and only if $\alpha>d/2-1$. In this case, if in addition the domain is $C^2$, we also establish spatial and temporal Holder regularity of the solution. When $d/2-1<\alpha<d/2$, we show that the Holder exponents are optimal and obtain exact local and uniform moduli of continuity, a Chung-type law of the iterated logarithm, and sharp small ball probability estimates for the solution.
We study the Neumann problem $u_t-\varepsilon\Delta u=e^u-1-au\ (a>1)$ on a smooth bounded domain $\Omega\subset\mathbb{R}^2$. For the spatially homogeneous problem, $0$ is stable, the positive equilibrium $\xi_a$ is unstable, and solutions starting above $\xi_a$ blow up in finite time. Although finite-time blow-up per...
We prove the optimal global regularity of admissible solutions to a transformed very fast diffusion equation in the range $-1<p<0$, posed on smooth bounded domains with zero Dirichlet boundary data and initial data comparable to the distance function. More precisely, we establish existence and uniqueness and show that...
Tian-Ling Jin, Xushan Tu, Jingang Xiong et al.· 0 citations
We establish interior maximal $L^{q_c}$-regularity for bounded strong solutions of $u_t-\Delta u+|Du|^\gamma=f$ in $\mathbb{T}^d\times(0,T)$, where $d\geq 2$, $\gamma>2$, and $q_c=(d+2)(\gamma-1)/\gamma$. The estimates are uniform for uniformly bounded families of solutions whose source terms range over a bounded, unif...
We consider the three-dimensional stochastic wave equation (SWE) driven by a multiplicative Gaussian noise that is white in time and colored in space: \[ \frac{\partial^2 u}{\partial t^2} = \Delta u + b\bigl(u\bigr) + \sigma\bigl(u\bigr)\,\dot{W}, \] where the drift function $ b $ and diffusion coefficient $\sigma$ are...
We study instantaneous shrinking of supports for nonnegative solutions of the stochastic partial differential equation \[ \partial_t u=a(t,x)\,\partial_x^2 u + b(t,x)\,\partial_x u + c(t,x)\,u +\sigma(u)\,\xi(t,x), \qquad (t,x)\in(0,\infty)\times\mathbb R, \] where $\xi$ is space-time white noise, the coefficients $a$,...
We establish sharp temporal lower bounds for the full spatial $L^2$-norm of energy solutions to $\partial_tu-\Delta u=vu$ on $\mathbb{R}^n$, with $n\geq3$. For global solutions, nonvanishing at a single time implies the lower bound $ce^{-Ct}$ when $v$ is small in the scale-invariant space $L^\infty_tL^{n/2}_x$, and $ce...
I. Kukavica, Qi Xu· 1 citation· ⚡1
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