Existence And Uniqueness Of Strong Solutions For A Class Of Stochastic Partial Differential Equations With Non-lipschitz Coefficients And Gradient

In recent years,stochastic partial differential equations,as an active research direction with rapid development,have been widely studied by experts at home and abroad.The study of stochastic partial differential equations involves many interdisciplinary,such as probability theory,stochastic analysis,partial differential equations and so on.The research background of stochastic partial differential equations originates from some applied disciplines such as physics and economics,therefore,the study of stochastic partial differential equations has both theoretical and practical significance.Most studies of stochastic partial differential equations are driven by white Gaussian noise and the coefficients of the equations satisfy the Lipschitz condition,but random phenomena in nature are complex,a stochastic partial differential equation driven by more complex noise and with the equation coefficients satisfying weaker conditions needs to be considered.At present,there are relatively few conclusions about the equation coefficients that are driven by Gaussian colored noise and satisfy the non-Lipschitz condition,the study of both the drift term and the diffusion term coefficients with gradients is a new challenge.Therefore,it is very necessary to study the stochastic partial differential equations of non-Lipschitz coefficients with gradients driven by Gaussian colored noise.In this dissertation,we study a stochastic partial differential equation driven by Gaussian colored noise,the existence,uniqueness,and moment estimation of a strong solution of a stochastic partial differential equation driven by Gaussian colored noise,where both the drift term and the diffusion term coefficients are accompanied by gradients,are considered.Firstly,the normalization method is used to obtain the only conclusion that the strong solution of the equation is unique;secondly,the strong solution of the equation is constructed recursively by using the transition semigroup,and then obtain the convergence of the strong solution by proving that the constructed strong solution is a Cauchy sequence,thereby,the existence of the strong solution is obtained;finally,we obtain the mild solution of the equation by controlling the convergence theorem,and use the Gronwall lemma to obtain the moment estimate of the solution.