We consider the well-posedness of a stochastic evolution problem in a bounded Lipschitz domain D ⊂ ℝd with homogeneous Dirichlet boundary conditions and an initial condition in L2(D). The main technical difficulties in proving the result of existence and uniqueness of a solution arise from the nonlinear diffusion-convection operator in divergence form which is given by the sum of a Carathéodory function satisfying p-type growth associated with coercivity assumptions and a Lipschitz continuous perturbation. In particular, we consider the case 1 < p < 2 with an appropriate lower bound on p determined by the space dimension. Another difficulty arises from the fact that the additive stochastic perturbation with values in L2(D) on the right-hand side of the equation does not inherit the Sobolev spatial regularity from the solution as in the multiplicative noise case.

中文翻译：

---

具有强连续扰动的非线性 SPDE 的适定性

我们考虑有界 Lipschitz 域中随机演化问题的适定性D⊂ℝd具有齐次狄利克雷边界条件和初始条件大号2(D）。证明解的存在性和唯一性结果的主要技术困难来自发散形式的非线性扩散-对流算子，该算子由满足的 Carathéodory 函数之和给出p型增长与矫顽力假设和 Lipschitz 连续扰动相关。特别是，我们考虑情况 1 <p< 2 具有适当的下限p由空间维度决定。另一个困难来自这样一个事实，即加性随机扰动与值大号2(D) 等式右侧的不从解中继承 Sobolev 空间规律性，如在乘性噪声情况下。