English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We present uniqueness and existence in weighted Sobolev spaces of theequation $$ u_t=(au_{xx}+bu_x+cu)+ \xi |u|^{1+\lambda} {\dot{B}}, \quad\,\,t>0, \, x\in (0,1) $$ with initial data $u(0,\cdot)=u_0$ and zero boundarydata. Here $\lambda\in [0,1/2)$, $\dot{B}$ is a space-time white noise, and thecoefficients $a,b,c$ and the function $\xi$ depend on $(\omega,t,x)$ and theinitial data $u_0$ depends on $(\omega,x)$.  More importantly, we obtain various interior H\"older regularities andboundary behaviors of the solution. For instance, if the initial data is inappropriate $L_p$ spaces, then for any small $\varepsilon>0$ and $T<\infty$,almost surely $$ \rho^{-1/2-\kappa}u \inC^{\frac{1}{4}-\frac{\kappa}{2}-\varepsilon,\frac{1}{2}-\kappa-\varepsilon}_{t,x}([0,T]\times (0,1)), \quad \forall\,\kappa\in (\lambda, 1/2), $$ where $\rho(x)$ is the distance from $x$ to theboundary. Taking $\kappa \downarrow \lambda$, one gets the the maximal H\"olderexponents in time and space, which are $1/4-\lambda/2-\varepsilon$ and$1/2-\lambda-\varepsilon $ respectively. Also, letting $\kappa \uparrow 1/2$,one gets better decay or behavior near the boundary.

Boundary behavior and interior H\"older regularity of solution to  nonlinear stochastic partial differential equations driven by space-time  white noise