This survey is concerned with the new developments on existence and uniqueness of solutions of some basic models in atmospheric dynamics, such as two-and three-dimensional quasi-geostrophic models and three-dimensiona...This survey is concerned with the new developments on existence and uniqueness of solutions of some basic models in atmospheric dynamics, such as two-and three-dimensional quasi-geostrophic models and three-dimensional balanced model. The main aim of this paper is to introduce some results about the global and local (with respect to time) existence of solutions given by the authors in recent years, but others' important contributions and the literature on this subject are also quoted. We discuss briefly the relationships among the existence and uniqueness, physical instability and computational instability. In the appendixes, some key mathematical techniques in obtaining our results are presented, which are of vital importance to other problems in geophysical fluid dynamics as well.展开更多
A fixed mesh variational formulation is used to establish existence and uniqueness of the solution of ordinary differential equations with (in finitely many) state-dependent in pulses on the right-hand side. This appr...A fixed mesh variational formulation is used to establish existence and uniqueness of the solution of ordinary differential equations with (in finitely many) state-dependent in pulses on the right-hand side. This approach gives a natural numerical scheme to approximate the solution.The convergence of the approximation is proved and its asymptatic order obtained.展开更多
This paper is concerned with the following second-order vector boundary value problem :x^R=f(t,Sx,x,x'),0〈t〈1,x(0)=A,g(x(1),x'(1))=B,where x,f,g,A and B are n-vectors. Under appropriate assumptions,exis...This paper is concerned with the following second-order vector boundary value problem :x^R=f(t,Sx,x,x'),0〈t〈1,x(0)=A,g(x(1),x'(1))=B,where x,f,g,A and B are n-vectors. Under appropriate assumptions,existence and uniqueness of solutions are obtained by using upper and lower solutions method.展开更多
In this paper, we study the long-term dynamic behavior of a class of generalized high-order Kirchhoff-type coupled wave equations. Firstly, the existence of uniqueness global solution of this kind of equations in E<...In this paper, we study the long-term dynamic behavior of a class of generalized high-order Kirchhoff-type coupled wave equations. Firstly, the existence of uniqueness global solution of this kind of equations in E<sub>k</sub> space is proved by prior estimation and Galerkin method;Then, through using Rellich-Kondrachov compact embedding theorem, it is proved that the solution semigroup S(t) has the family of the global attractors A<sub>k</sub> in space E<sub>k</sub>;Finally, through linearization method, proves that the operator semigroup S(t) Frechet differentiable and the attenuation of linearization problem volume element. Furthermore, we can obtain the finite Hausdorff dimension and Fractal dimension of the family of the global attractors A<sub>k</sub>.展开更多
In this paper, we study the long time behavior of a class of Kirchhoff equations with high order strong dissipative terms. On the basis of the proper hypothesis of rigid term and nonlinear term of Kirchhoff equation, ...In this paper, we study the long time behavior of a class of Kirchhoff equations with high order strong dissipative terms. On the basis of the proper hypothesis of rigid term and nonlinear term of Kirchhoff equation, firstly, we evaluate the equation via prior estimate in the space <em>E</em><sub>0</sub> and <em>E<sub>k</sub></em>, and verify the existence and uniqueness of the solution of the equation by using Galerkin’s method. Then, we obtain the bounded absorptive set <em>B</em><sub><em>0k</em> </sub>on the basis of the prior estimate. Moreover, by using the Rellich-Kondrachov Compact Embedding theorem, we prove that the solution semigroup <em>S</em>(<em>t</em>) of the equation has the family of the global attractor <em>A<sub>k</sub></em><sub> </sub>in space <em>E<sub>k</sub></em>. Finally, we prove that the solution semigroup <em>S</em>(<em>t</em>) is Frechet differentiable on <em>E<sub>k</sub></em> via linearizing the equation. Furthermore, we can obtain the finite Hausdorff dimension and Fractal dimension of the family of the global attractor <em>A<sub>k</sub></em>.展开更多
In this paper, we studied the long-time properties of solutions of generalized Kirchhoff-type equation with strongly damped terms. Firstly, appropriate assumptions are made for the nonlinear source term <span style...In this paper, we studied the long-time properties of solutions of generalized Kirchhoff-type equation with strongly damped terms. Firstly, appropriate assumptions are made for the nonlinear source term <span style="white-space:nowrap;"><em>g</em> (<em>u</em>)</span> and Kirchhoff stress term <span style="white-space:nowrap;"><em>M</em> (<em>s</em>)</span> in the equation, and the existence and uniqueness of the solution are proved by using uniform prior estimates of time and Galerkin’s finite element method. Then, abounded absorption set <em>B</em><sub>0<em>k</em></sub> is obtained by prior estimation, and the Rellich-kondrachov’s compact embedding theorem is used to prove that the solution semigroup <span style="white-space:nowrap;"><em>S</em> (<em>t</em>)</span> generated by the equation has a family of the global attractor <span style="white-space:nowrap;"><em>A</em><sub><em>k</em></sub></span> in the phase space <img src="Edit_250265b5-40f0-4b6c-b669-958eb1938010.png" width="120" height="20" alt="" />. Finally, linearize the equation and verify that the semigroups are Frechet diifferentiable on <em>E<sub>k</sub></em>. Then, the upper boundary estimation of the Hausdorff dimension and Fractal dimension of a family of the global attractor <em>A<sub>k</sub></em> was obtained.展开更多
In this article, the existence, uniqueness and regularities of the global generalized solution and global classical solution for the periodic boundary value problem and the Cauchy problem of the general cubic double d...In this article, the existence, uniqueness and regularities of the global generalized solution and global classical solution for the periodic boundary value problem and the Cauchy problem of the general cubic double dispersion equationutt - uxx - auxxtt + bux4 - duxxt = f(u)xxare proved, and the sufficient conditions of blow-up of the solutions for the Cauchy problems in finite time are given.展开更多
In this paper, we consider a class of generalized nonlinear Kirchhoff-Sine-Gordon equation . By a priori estimation, we first prove the existence and uniqueness of solutions to the initial boundary value conditio...In this paper, we consider a class of generalized nonlinear Kirchhoff-Sine-Gordon equation . By a priori estimation, we first prove the existence and uniqueness of solutions to the initial boundary value conditions, and then we study the global attractors of the equation.展开更多
Illicit drug use is a significant problem that causes great material and moral losses and threatens the future of the society.For this reason,illicit drug use and related crimes are the most significant criminal cases...Illicit drug use is a significant problem that causes great material and moral losses and threatens the future of the society.For this reason,illicit drug use and related crimes are the most significant criminal cases examined by scientists.This paper aims at modeling the illegal drug use using the Atangana-Baleanu fractional derivative with Mittag-Leffler kernel.Also,in this work,the existence and uniqueness of solutions of the fractional-order Illicit drug use model are discussed via Picard-Lindelöf theorem which provides successive approximations using a convergent sequence.Then the stability analysis for both disease-free and endemic equilibrium states is conducted.A numerical scheme based on the known Adams-Bashforth method is designed in fractional form to approximate the novel Atangana-Baleanu fractional operator of order 0<a≤1.Finally,numerical simulation results based on different values of fractional order,which also serve as control parameter,are presented to justify the theoretical findings.展开更多
Some basic problems on the level set methods were discussed, such as the method used to preserve the distance junction , the existence and uniqueness of solution for the level set equations. The main contribution is t...Some basic problems on the level set methods were discussed, such as the method used to preserve the distance junction , the existence and uniqueness of solution for the level set equations. The main contribution is to prove that in a neighborhood of the initial zero level set, the level set equations with the restriction of the distance function have a unique solution, which must be the signed distance function with respect to the evolving surface. Some skillful approaches were used: Noticing that any solution for the original equation was a distance function, the original level set equations were transformed into a simpler alternative form. Moreover, since the new system was not a classical one, the system was transformed into an ordinary one, for which the implicit function method was adopted.展开更多
Existence and uniqueness conditions for nonnegative solutions to initial value prob- lems of general sublinear-linear differential equations are obtained.They extend the uniqueness theorem due to H.Murakami~[6] and th...Existence and uniqueness conditions for nonnegative solutions to initial value prob- lems of general sublinear-linear differential equations are obtained.They extend the uniqueness theorem due to H.Murakami~[6] and the main results of H.G.Kaper and M.K.Kwong~[4].展开更多
The existence and uniqueness of the state for 2 × 2 Dirichlet cooperative elliptic systems under conjugation conditions are proved using Lax-Milgram lemma, then the boundary control for these systems is discussed...The existence and uniqueness of the state for 2 × 2 Dirichlet cooperative elliptic systems under conjugation conditions are proved using Lax-Milgram lemma, then the boundary control for these systems is discussed. The set of equations and inequalities that characterizes this boundary control is found by theory of Lions, Sergienko and Deineka. The problem for cooperative Neumann elliptic systems under conjugation conditions is also considered. Finally, the problem for <em>n</em> × <em>n</em> cooperative elliptic systems under conjugation conditions is established.展开更多
In this paper, we study the initial boundary value problem of coupled generalized Kirchhoff equations. Firstly, the rigid term and nonlinear term of Kirchhoff equation are assumed appropriately to obtain the prior est...In this paper, we study the initial boundary value problem of coupled generalized Kirchhoff equations. Firstly, the rigid term and nonlinear term of Kirchhoff equation are assumed appropriately to obtain the prior estimates of the equation in E<sub>0</sub> and E<sub>k</sub> space, and then the existence and uniqueness of solution is verified by Galerkin’s method. Then, the solution semigroup S(t) is defined, and the bounded absorptive set B<sub>k</sub> is obtained on the basis of prior estimation. Through using Rellich-Kondrachov compact embedding theorem, it is proved that the solution semigroup S(t) has the family of the global attractors A<sub>k</sub> in space E<sub>k</sub>. Finally, by linearizing the equation, it is proved that the solution semigroup S(t) is Frechet differentiable on E<sub>k</sub>, and the family of global attractors A<sub>k</sub> have finite Hausdroff dimension and Fractal dimension.展开更多
This paper mainly studies the initial value problems of Kirchhoff-type coupled equations. Firstly, by giving the hypothesis of Kirchhoff stress term , the Galerkin’s method obtains the existence uniqueness of the ove...This paper mainly studies the initial value problems of Kirchhoff-type coupled equations. Firstly, by giving the hypothesis of Kirchhoff stress term , the Galerkin’s method obtains the existence uniqueness of the overall solution of the above problem by using a priori estimates in the spaces of E<sub>0</sub> and E<sub>k</sub>, and secondly, it proves that there is a family of global attractors for the above problem, and finally estimates the Hausdorff dimension and the Fractal dimension of the family of global attractors.展开更多
In [1] and [2], the authors made a deep qualitative analysis of the equationwith the character of tangent detected phase and they mathematically provided atheoretical basis of why the phase looked loop has no look--lo...In [1] and [2], the authors made a deep qualitative analysis of the equationwith the character of tangent detected phase and they mathematically provided atheoretical basis of why the phase looked loop has no look--losing point. However,according to many practical experts, it is rather difficult to put such a phaselooked loop into practice, though it has fine properties. W. C. Lindsey [3] made a展开更多
This paper deals with the uniqueness and existence of periodic solutions of neutral Volterraintegrodifferential equations (1) and (2). Some new unique existence criteria are obtained.
For the image of a smooth surface object fully contained within the field of view and illuminated in an arbitrary direction, this paper discusses the ekistence and uniqueness of the conditions for solving a shape-from...For the image of a smooth surface object fully contained within the field of view and illuminated in an arbitrary direction, this paper discusses the ekistence and uniqueness of the conditions for solving a shape-from-shading problem under the conditions that the Fourier series expansion of the image intensity contains only zero and first order terms in a polar coordinate system. Three theorems are established, one for the ekistence and two for the uniqueness of z-axis symmetric shape from shading.展开更多
We study the solvability of the Cauchy problem (1.1)-(1.2) for the largest possible class of initial values,for which (1.1)-(1.2) has a local solution.Moreover,we also study the critical case related to the in...We study the solvability of the Cauchy problem (1.1)-(1.2) for the largest possible class of initial values,for which (1.1)-(1.2) has a local solution.Moreover,we also study the critical case related to the initial value u<sub>0</sub>,for 1【p【∞.展开更多
文摘This survey is concerned with the new developments on existence and uniqueness of solutions of some basic models in atmospheric dynamics, such as two-and three-dimensional quasi-geostrophic models and three-dimensional balanced model. The main aim of this paper is to introduce some results about the global and local (with respect to time) existence of solutions given by the authors in recent years, but others' important contributions and the literature on this subject are also quoted. We discuss briefly the relationships among the existence and uniqueness, physical instability and computational instability. In the appendixes, some key mathematical techniques in obtaining our results are presented, which are of vital importance to other problems in geophysical fluid dynamics as well.
文摘A fixed mesh variational formulation is used to establish existence and uniqueness of the solution of ordinary differential equations with (in finitely many) state-dependent in pulses on the right-hand side. This approach gives a natural numerical scheme to approximate the solution.The convergence of the approximation is proved and its asymptatic order obtained.
文摘This paper is concerned with the following second-order vector boundary value problem :x^R=f(t,Sx,x,x'),0〈t〈1,x(0)=A,g(x(1),x'(1))=B,where x,f,g,A and B are n-vectors. Under appropriate assumptions,existence and uniqueness of solutions are obtained by using upper and lower solutions method.
文摘In this paper, we study the long-term dynamic behavior of a class of generalized high-order Kirchhoff-type coupled wave equations. Firstly, the existence of uniqueness global solution of this kind of equations in E<sub>k</sub> space is proved by prior estimation and Galerkin method;Then, through using Rellich-Kondrachov compact embedding theorem, it is proved that the solution semigroup S(t) has the family of the global attractors A<sub>k</sub> in space E<sub>k</sub>;Finally, through linearization method, proves that the operator semigroup S(t) Frechet differentiable and the attenuation of linearization problem volume element. Furthermore, we can obtain the finite Hausdorff dimension and Fractal dimension of the family of the global attractors A<sub>k</sub>.
文摘In this paper, we study the long time behavior of a class of Kirchhoff equations with high order strong dissipative terms. On the basis of the proper hypothesis of rigid term and nonlinear term of Kirchhoff equation, firstly, we evaluate the equation via prior estimate in the space <em>E</em><sub>0</sub> and <em>E<sub>k</sub></em>, and verify the existence and uniqueness of the solution of the equation by using Galerkin’s method. Then, we obtain the bounded absorptive set <em>B</em><sub><em>0k</em> </sub>on the basis of the prior estimate. Moreover, by using the Rellich-Kondrachov Compact Embedding theorem, we prove that the solution semigroup <em>S</em>(<em>t</em>) of the equation has the family of the global attractor <em>A<sub>k</sub></em><sub> </sub>in space <em>E<sub>k</sub></em>. Finally, we prove that the solution semigroup <em>S</em>(<em>t</em>) is Frechet differentiable on <em>E<sub>k</sub></em> via linearizing the equation. Furthermore, we can obtain the finite Hausdorff dimension and Fractal dimension of the family of the global attractor <em>A<sub>k</sub></em>.
文摘In this paper, we studied the long-time properties of solutions of generalized Kirchhoff-type equation with strongly damped terms. Firstly, appropriate assumptions are made for the nonlinear source term <span style="white-space:nowrap;"><em>g</em> (<em>u</em>)</span> and Kirchhoff stress term <span style="white-space:nowrap;"><em>M</em> (<em>s</em>)</span> in the equation, and the existence and uniqueness of the solution are proved by using uniform prior estimates of time and Galerkin’s finite element method. Then, abounded absorption set <em>B</em><sub>0<em>k</em></sub> is obtained by prior estimation, and the Rellich-kondrachov’s compact embedding theorem is used to prove that the solution semigroup <span style="white-space:nowrap;"><em>S</em> (<em>t</em>)</span> generated by the equation has a family of the global attractor <span style="white-space:nowrap;"><em>A</em><sub><em>k</em></sub></span> in the phase space <img src="Edit_250265b5-40f0-4b6c-b669-958eb1938010.png" width="120" height="20" alt="" />. Finally, linearize the equation and verify that the semigroups are Frechet diifferentiable on <em>E<sub>k</sub></em>. Then, the upper boundary estimation of the Hausdorff dimension and Fractal dimension of a family of the global attractor <em>A<sub>k</sub></em> was obtained.
文摘In this article, the existence, uniqueness and regularities of the global generalized solution and global classical solution for the periodic boundary value problem and the Cauchy problem of the general cubic double dispersion equationutt - uxx - auxxtt + bux4 - duxxt = f(u)xxare proved, and the sufficient conditions of blow-up of the solutions for the Cauchy problems in finite time are given.
文摘In this paper, we consider a class of generalized nonlinear Kirchhoff-Sine-Gordon equation . By a priori estimation, we first prove the existence and uniqueness of solutions to the initial boundary value conditions, and then we study the global attractors of the equation.
文摘Illicit drug use is a significant problem that causes great material and moral losses and threatens the future of the society.For this reason,illicit drug use and related crimes are the most significant criminal cases examined by scientists.This paper aims at modeling the illegal drug use using the Atangana-Baleanu fractional derivative with Mittag-Leffler kernel.Also,in this work,the existence and uniqueness of solutions of the fractional-order Illicit drug use model are discussed via Picard-Lindelöf theorem which provides successive approximations using a convergent sequence.Then the stability analysis for both disease-free and endemic equilibrium states is conducted.A numerical scheme based on the known Adams-Bashforth method is designed in fractional form to approximate the novel Atangana-Baleanu fractional operator of order 0<a≤1.Finally,numerical simulation results based on different values of fractional order,which also serve as control parameter,are presented to justify the theoretical findings.
基金the National Natural Science Foundation of China (6001161942, 60203003)
文摘Some basic problems on the level set methods were discussed, such as the method used to preserve the distance junction , the existence and uniqueness of solution for the level set equations. The main contribution is to prove that in a neighborhood of the initial zero level set, the level set equations with the restriction of the distance function have a unique solution, which must be the signed distance function with respect to the evolving surface. Some skillful approaches were used: Noticing that any solution for the original equation was a distance function, the original level set equations were transformed into a simpler alternative form. Moreover, since the new system was not a classical one, the system was transformed into an ordinary one, for which the implicit function method was adopted.
文摘Existence and uniqueness conditions for nonnegative solutions to initial value prob- lems of general sublinear-linear differential equations are obtained.They extend the uniqueness theorem due to H.Murakami~[6] and the main results of H.G.Kaper and M.K.Kwong~[4].
文摘The existence and uniqueness of the state for 2 × 2 Dirichlet cooperative elliptic systems under conjugation conditions are proved using Lax-Milgram lemma, then the boundary control for these systems is discussed. The set of equations and inequalities that characterizes this boundary control is found by theory of Lions, Sergienko and Deineka. The problem for cooperative Neumann elliptic systems under conjugation conditions is also considered. Finally, the problem for <em>n</em> × <em>n</em> cooperative elliptic systems under conjugation conditions is established.
文摘In this paper, we study the initial boundary value problem of coupled generalized Kirchhoff equations. Firstly, the rigid term and nonlinear term of Kirchhoff equation are assumed appropriately to obtain the prior estimates of the equation in E<sub>0</sub> and E<sub>k</sub> space, and then the existence and uniqueness of solution is verified by Galerkin’s method. Then, the solution semigroup S(t) is defined, and the bounded absorptive set B<sub>k</sub> is obtained on the basis of prior estimation. Through using Rellich-Kondrachov compact embedding theorem, it is proved that the solution semigroup S(t) has the family of the global attractors A<sub>k</sub> in space E<sub>k</sub>. Finally, by linearizing the equation, it is proved that the solution semigroup S(t) is Frechet differentiable on E<sub>k</sub>, and the family of global attractors A<sub>k</sub> have finite Hausdroff dimension and Fractal dimension.
文摘This paper mainly studies the initial value problems of Kirchhoff-type coupled equations. Firstly, by giving the hypothesis of Kirchhoff stress term , the Galerkin’s method obtains the existence uniqueness of the overall solution of the above problem by using a priori estimates in the spaces of E<sub>0</sub> and E<sub>k</sub>, and secondly, it proves that there is a family of global attractors for the above problem, and finally estimates the Hausdorff dimension and the Fractal dimension of the family of global attractors.
文摘In [1] and [2], the authors made a deep qualitative analysis of the equationwith the character of tangent detected phase and they mathematically provided atheoretical basis of why the phase looked loop has no look--losing point. However,according to many practical experts, it is rather difficult to put such a phaselooked loop into practice, though it has fine properties. W. C. Lindsey [3] made a
基金project is supported by National Natural Science Foundation of China
文摘This paper deals with the uniqueness and existence of periodic solutions of neutral Volterraintegrodifferential equations (1) and (2). Some new unique existence criteria are obtained.
文摘For the image of a smooth surface object fully contained within the field of view and illuminated in an arbitrary direction, this paper discusses the ekistence and uniqueness of the conditions for solving a shape-from-shading problem under the conditions that the Fourier series expansion of the image intensity contains only zero and first order terms in a polar coordinate system. Three theorems are established, one for the ekistence and two for the uniqueness of z-axis symmetric shape from shading.
基金Project supported by the National Natural Science Foundation of China (19971070)
文摘We study the solvability of the Cauchy problem (1.1)-(1.2) for the largest possible class of initial values,for which (1.1)-(1.2) has a local solution.Moreover,we also study the critical case related to the initial value u<sub>0</sub>,for 1【p【∞.