In this paper,we consider the fourth-order parabolic equation with p(x)Laplacian and variable exponent source ut+∆^(2)u−div(|■u|^(p(x)−2■u))=|u|^(q(x))−1u.By applying potential well method,we obtain global existence...In this paper,we consider the fourth-order parabolic equation with p(x)Laplacian and variable exponent source ut+∆^(2)u−div(|■u|^(p(x)−2■u))=|u|^(q(x))−1u.By applying potential well method,we obtain global existence,asymptotic behavior and blow-up of solutions with initial energy J(u_(0))≤d.Moreover,we estimate the upper bound of the blow-up time for J(u_(0))≤0.展开更多
This paper deals with the global boundedness of a two-competing-species chemotaxis model with indirect signal production in a three-dimensional bounded domain.The current work extends prior results by ZHENG et al.(202...This paper deals with the global boundedness of a two-competing-species chemotaxis model with indirect signal production in a three-dimensional bounded domain.The current work extends prior results by ZHENG et al.(2022)who established global existence and boundedness of classical solution under the parameter constraintsµ_(1)µ_(2)a_(2)≥χ1(4+µ_(2)^(2)a _(2)^(2)),µ_(1)µ_(2)a_(1)≥χ2(4+µ_(1)^(2)a_(1)^( 2)).Our main contribution demonstrates that these technical conditions can be significantly relaxed toµ1≥5χ_(1),µ2≥5χ_(2),thereby expanding the admissible parameter space for solution boundedness.展开更多
Monge–Ampere equations(MAEs)are fully nonlinear second-order partial differential equations(PDEs),which are closely related to various fields including optimal transport(OT)theory,geometrical optics and affine geomet...Monge–Ampere equations(MAEs)are fully nonlinear second-order partial differential equations(PDEs),which are closely related to various fields including optimal transport(OT)theory,geometrical optics and affine geometry.Despite their significance,MAEs are extremely challenging to solve.Although some classical numerical approaches can solve MAEs,their computational efficiency deteriorates significantly on fine grids,with convergence often heavily dependent on the quality of initial estimate.Research on deep learning methods for solving MAEs is still in its early stages,which predominantly addresses simple formulations with basic Dirichlet boundary conditions.Here,we propose a deep learning method based on physicsdriven deep neural networks,enabling the solution of both simple and generalised MAEs with transport boundary conditions.In this method,we deal with two first-order sub-equations separated from MAE instead of solving the single MAE directly,which facilitates the imposition of transport boundary conditions and simplifies the training of neural networks.Moreover,we constrain the convexity of solution using the Lagrange multiplier method and maintain the optimisation process differentiable with bilinear interpolation.We provide three progressively complex examples ranging from a simple MAE with an analytical solution to a highly nonlinear variant arising in phase retrieval to validate the effectiveness of our method.For comparison,we benchmark against state-of-the-art deep learning approaches that have been systematically adapted to accommodate the specific requirements of each example.展开更多
The paper considers the initial value problem of inhomogeneous fourth-order Schr¨odinger equation with potential in energy space H^(2)(R^(d)).The global well-posedness is obtained in dimensions d≥5 resorting to ...The paper considers the initial value problem of inhomogeneous fourth-order Schr¨odinger equation with potential in energy space H^(2)(R^(d)).The global well-posedness is obtained in dimensions d≥5 resorting to contractive mapping principle,Strichartz estimates,Caffarelli-Kohn-Nirenberg-type inequality and the continuity method.展开更多
基金Supported by NSFC(No.12101482)the Natural Science Foundation of Shaanxi Province,China(No.2018JQ1052)。
文摘In this paper,we consider the fourth-order parabolic equation with p(x)Laplacian and variable exponent source ut+∆^(2)u−div(|■u|^(p(x)−2■u))=|u|^(q(x))−1u.By applying potential well method,we obtain global existence,asymptotic behavior and blow-up of solutions with initial energy J(u_(0))≤d.Moreover,we estimate the upper bound of the blow-up time for J(u_(0))≤0.
基金Supported by the National Natural Science Foundation of China(12301631)the Natural Science Foundation of Qinghai Province(2023-ZJ-949Q)。
文摘This paper deals with the global boundedness of a two-competing-species chemotaxis model with indirect signal production in a three-dimensional bounded domain.The current work extends prior results by ZHENG et al.(2022)who established global existence and boundedness of classical solution under the parameter constraintsµ_(1)µ_(2)a_(2)≥χ1(4+µ_(2)^(2)a _(2)^(2)),µ_(1)µ_(2)a_(1)≥χ2(4+µ_(1)^(2)a_(1)^( 2)).Our main contribution demonstrates that these technical conditions can be significantly relaxed toµ1≥5χ_(1),µ2≥5χ_(2),thereby expanding the admissible parameter space for solution boundedness.
基金supported by CAAI-Huawei MindSpore Open Fund(CAAIXSJLJJ-2022-010A).
文摘Monge–Ampere equations(MAEs)are fully nonlinear second-order partial differential equations(PDEs),which are closely related to various fields including optimal transport(OT)theory,geometrical optics and affine geometry.Despite their significance,MAEs are extremely challenging to solve.Although some classical numerical approaches can solve MAEs,their computational efficiency deteriorates significantly on fine grids,with convergence often heavily dependent on the quality of initial estimate.Research on deep learning methods for solving MAEs is still in its early stages,which predominantly addresses simple formulations with basic Dirichlet boundary conditions.Here,we propose a deep learning method based on physicsdriven deep neural networks,enabling the solution of both simple and generalised MAEs with transport boundary conditions.In this method,we deal with two first-order sub-equations separated from MAE instead of solving the single MAE directly,which facilitates the imposition of transport boundary conditions and simplifies the training of neural networks.Moreover,we constrain the convexity of solution using the Lagrange multiplier method and maintain the optimisation process differentiable with bilinear interpolation.We provide three progressively complex examples ranging from a simple MAE with an analytical solution to a highly nonlinear variant arising in phase retrieval to validate the effectiveness of our method.For comparison,we benchmark against state-of-the-art deep learning approaches that have been systematically adapted to accommodate the specific requirements of each example.
基金Supported by National Natural Science Foundation of China(Grant No.11601122).
文摘The paper considers the initial value problem of inhomogeneous fourth-order Schr¨odinger equation with potential in energy space H^(2)(R^(d)).The global well-posedness is obtained in dimensions d≥5 resorting to contractive mapping principle,Strichartz estimates,Caffarelli-Kohn-Nirenberg-type inequality and the continuity method.
