\(L_p\)-solvability and Hölder regularity for stochastic time fractional Burgers’ equations driven by multiplicative space-time white noise

This paper establishes \(L_p\) $L_p$ -solvability for stochastic time fractional Burgers’ equations driven by multiplicative space-time white noise: \(\begin{aligned} \partial _t^\alpha u = a^{ij}u_{x^ix^j} + b^{i}u_{x^i} + cu + {\bar{b}}^i u u_{x^i} + \partial _t^\beta \int _0^t \sigma (u)dW_t,\quad t>0;\quad u(0,\cdot ) = u_0, \end{aligned}\) $$\begin{array}{c}\hfill {\partial }_t^{\alpha }u=a^{\mathit{ij}}{u}_{x^i{x}^j}+b^i{u}_{x^i}+cu+{\overline{b}}^{i}u{u}_{x^i}+{\partial }_t^{\beta }{\int }_0^t\sigma (u)d{W}_t,t>0;u(0,·)=u_0,\end{array}$$ where \(\alpha \in (0,1)\) $\alpha \in (0,1)$ , \(\beta < 3\alpha /4+1/2\) $\beta <3\alpha /4+1/2$ , and \(d< 4--2(2\beta -1)_+/\alpha \) $d<4--2{(2\beta -1)}_{+}/\alpha$ . The operators \(\partial _t^\alpha \) ${\partial }_t^{\alpha }$ and \(\partial _t^\beta \) ${\partial }_t^{\beta }$ are the Caputo fractional derivatives of order \(\alpha \) $\alpha$ and \(\beta \) $\beta$ , respectively. The process \(W_t\) $W_t$ is an \(L_2(\mathbb {R}^d)\) $L_2({\mathbb{R}}^d)$ -valued cylindrical Wiener process, and the coefficients \(a^{ij}, b^i, c, {\bar{b}}^{i}\) $a^{\mathit{ij}},b^i,c,{\overline{b}}^i$ and \(\sigma (u)\) $\sigma (u)$ are random. In addition to the uniqueness and existence of a solution, the Hölder regularity of the solution is also established. For example, for any constant \(T<\infty \) $T<\infty$ , small \(\varepsilon >0\) $\epsilon >0$ , and almost sure \(\omega \in \varOmega \) $\omega \in \Omega$ , \(\begin{aligned} \sup _{x\in \mathbb {R}^d}|u(\omega ,\cdot ,x)|_{C^{\left[ \frac{\alpha }{2}\left( \left( 2-(2\beta -1)_+/\alpha -d/2 \right) \wedge 1 \right) +\frac{(2\beta -1)_{-}}{2} \right] \wedge 1-\varepsilon }([0,T])}<\infty \end{aligned}\) $$\begin{array}{c}\hfill \underset{x\in {\mathbb{R}}^d}{sup}{|u(\omega ,·,x)|}_{C^{\left[\frac{\alpha }{2},\left(\left(2,-,{(2\beta -1)}_{+},/,\alpha ,-,d,/,2\right),\wedge ,1\right),+,\frac{{(2\beta -1)}_{-}}{2}\right]\wedge 1-\epsilon }([0,T])}<\infty \end{array}$$ and \(\begin{aligned} \sup _{t\le T}|u(\omega ,t,\cdot )|_{C^{\left( 2-(2\beta -1)_+/\alpha -d/2 \right) \wedge 1 - \varepsilon }(\mathbb {R}^d)} < \infty . \end{aligned}\) $$\begin{array}{c}\hfill \underset{t\le T}{sup}{|u(\omega ,t,·)|}_{C^{\left(2,-,{(2\beta -1)}_{+},/,\alpha ,-,d,/,2\right)\wedge 1-\epsilon }({\mathbb{R}}^d)}<\infty .\end{array}$$ The Hölder regularity of the solution in time changes behavior at \(\beta = 1/2\) $\beta =1/2$ . Furthermore, if \(\beta \ge 1/2\) $\beta \ge 1/2$ , then the Hölder regularity of the solution in time is \(\alpha /2\) $\alpha /2$ times that in space.