In this paper,we establish global classical solutions of semilinear wave equations with small compact supported initial data posed on the product space R^(3)×T.The semilinear nonlinearity is assumed to be of the ...In this paper,we establish global classical solutions of semilinear wave equations with small compact supported initial data posed on the product space R^(3)×T.The semilinear nonlinearity is assumed to be of the cubic form.The main ingredient here is the establishment of the L^(2)-L^(∞)decay estimates and the energy estimates for the linear problem,which are adapted to the wave equation on the product space.The proof is based on the Fourier mode decomposition of the solution with respect to the periodic direction,the scaling technique,and the combination of the decay estimates and the energy estimates.展开更多
In this paper, we investigate a blow-up phenomenon for a semilinear parabolic system on locally finite graphs. Under some appropriate assumptions on the curvature condition CDE’(n,0), the polynomial volume growth of ...In this paper, we investigate a blow-up phenomenon for a semilinear parabolic system on locally finite graphs. Under some appropriate assumptions on the curvature condition CDE’(n,0), the polynomial volume growth of degree m, the initial values, and the exponents in absorption terms, we prove that every non-negative solution of the semilinear parabolic system blows up in a finite time. Our current work extends the results achieved by Lin and Wu (Calc Var Partial Differ Equ, 2017, 56: Art 102) and Wu (Rev R Acad Cien Serie A Mat, 2021, 115: Art 133).展开更多
In this paper, we present the a posteriori error estimate of two-grid mixed finite element methods by averaging techniques for semilinear elliptic equations. We first propose the two-grid algorithms to linearize the m...In this paper, we present the a posteriori error estimate of two-grid mixed finite element methods by averaging techniques for semilinear elliptic equations. We first propose the two-grid algorithms to linearize the mixed method equations. Then, the averaging technique is used to construct the a posteriori error estimates of the two-grid mixed finite element method and theoretical analysis are given for the error estimators. Finally, we give some numerical examples to verify the reliability and efficiency of the a posteriori error estimator.展开更多
In this paper an existence and uniqueness theorem of positive solutions to a class of semilinear elliptic systems is proved. Also, a necessary condition for the existence of the positive solution is obtained. As the a...In this paper an existence and uniqueness theorem of positive solutions to a class of semilinear elliptic systems is proved. Also, a necessary condition for the existence of the positive solution is obtained. As the application of the main theorem, two examples are given.展开更多
An optimization theoretic approach of coefficients in semilinear parabolic equation is presented. It is based on convex analysis techniques. General theorems on existence are proved in L1 setting. A necessary conditio...An optimization theoretic approach of coefficients in semilinear parabolic equation is presented. It is based on convex analysis techniques. General theorems on existence are proved in L1 setting. A necessary condition is given for the solutions of the parameter estimatioll problem.展开更多
This paper considers the initial and boundary value problem of some linear and semilinear Schrodinger equation with real potential and H01 initial data. The author obtains the homogenization of linear and semilinear S...This paper considers the initial and boundary value problem of some linear and semilinear Schrodinger equation with real potential and H01 initial data. The author obtains the homogenization of linear and semilinear Schrodinger equations and gives correctors for the homogenization of linear and semilinear Schrodinger equations.展开更多
An existence theorem is obtained for nonzero W^1,2(R^N) solutions of the following equations on R^N.-△u+ b(x)u = f(x,u), x∈R^N,where b is periodic for some variables and coercive for the others, f is superlinear.
This paper is concerned with the existence and the nodal character of the nontrivial solutions in the entire space for some semilinear elliptic equation involving critical Sobolev exponent.
The existence and uniqueness of singular solutions decaying like r m(see(1.4)) of the equation u + k i=1 ci|x|li upi = 0,x ∈ Rn(0.1) are obtained,where n ≥ 3,ci > 0,li > 2,i = 1,2,...,k,pi > 1,i = 1,2,...,k...The existence and uniqueness of singular solutions decaying like r m(see(1.4)) of the equation u + k i=1 ci|x|li upi = 0,x ∈ Rn(0.1) are obtained,where n ≥ 3,ci > 0,li > 2,i = 1,2,...,k,pi > 1,i = 1,2,...,k and the separation structure of singular solutions decaying like r(n 2) of eq.(0.1) are discussed.moreover,we obtain the explicit critical exponent ps(l)(see(1.9)).展开更多
The singularly perturbed boundary value problems for the semilinear elliptic equation are considered. Under suitable conditions and by using the fixed point theorem, the existence, uniqueness and asymptotic behavior o...The singularly perturbed boundary value problems for the semilinear elliptic equation are considered. Under suitable conditions and by using the fixed point theorem, the existence, uniqueness and asymptotic behavior of solution for the boundary value problems are studied.展开更多
We discussed the Dirichlet problem of semilinear elliptic equation (Pβ,ε): -β2△u=up+εe,u>0, in Ω ;u=0, on Ω, where ΩcR"(N≤4) is smooth and bounded domain, p= β,ε>0. We have proved that there exis...We discussed the Dirichlet problem of semilinear elliptic equation (Pβ,ε): -β2△u=up+εe,u>0, in Ω ;u=0, on Ω, where ΩcR"(N≤4) is smooth and bounded domain, p= β,ε>0. We have proved that there exist positive ε0. and ε1, such that when 0≤ε≤ε0, β>ε1, (Pβ,ε) has a single-peaked solution uβ,ε; furthermore, | uβ,ε|2-0 in the sense of measure as ε-0 and β-0.展开更多
In this paper, we study a semilinear Timoshenko system with heat conduction having two damping effects. The observation that two damping effects might lead to smaller decay rates for solutions in comparison to one dam...In this paper, we study a semilinear Timoshenko system with heat conduction having two damping effects. The observation that two damping effects might lead to smaller decay rates for solutions in comparison to one damping effect is rigorously proved here in providing optimality results. Moreover, the global well-posedness for small data is presented.展开更多
The numerical solution of a singularly perturbed problem for the semilinear parabolicdifferential equation with parabolic boundary layers is discussed.A nonlinear two-leveldifference scheme is constructed on the speci...The numerical solution of a singularly perturbed problem for the semilinear parabolicdifferential equation with parabolic boundary layers is discussed.A nonlinear two-leveldifference scheme is constructed on the special non-uniform grids.The uniform convergenceof this scheme is proved and some numerical examples are given.展开更多
This paper deals with the blow-up properties of solutions to semilinear heat equation ut-uxx= up in (0, 1) × (0, T) with the Neumann boundary condition ux(0, t) = 0, u:x1, t) = 1 on [0, T). The necessary and suff...This paper deals with the blow-up properties of solutions to semilinear heat equation ut-uxx= up in (0, 1) × (0, T) with the Neumann boundary condition ux(0, t) = 0, u:x1, t) = 1 on [0, T). The necessary and sufficient conditions under which all solutions to have a finite time blow-up and the exact blow-up rates are established. It is proved that the blow-up will occur only at the boundary x = 1. The asymptotic behavior near the blow-up time is also studied.展开更多
In this paper, we will consider following initial value problem of semilinear stochastic evolution equation in Hilbert Space: where W（t） is a wiener process in H, H and Y are two real separable Hilbert Spaces, A is ...In this paper, we will consider following initial value problem of semilinear stochastic evolution equation in Hilbert Space: where W（t） is a wiener process in H, H and Y are two real separable Hilbert Spaces, A is an infinitesimal generator of a strongly continuous semigroup s（t） on Y,f（t, y）: [0, T]×Y→Y, and G（t, y）: [0, T]×Y→L（H, Y）, y<sub>0</sub>: Ω→Y is a ramdom variable of square integrable. We apply theory of the semigroup and obtain two conclusions of uniqueness of the mild solution of （1） which include the corresponding results in [4].展开更多
In the studies of nonlinear partial differential equations, the influence, from the singularities of coefficients to the singularities of solution, is a field that has not been dealt with. In this paper, we discuss a ...In the studies of nonlinear partial differential equations, the influence, from the singularities of coefficients to the singularities of solution, is a field that has not been dealt with. In this paper, we discuss a simple case of semilinear equations under the frame of the space of conormal distributions. We prove the result that the solution has the same singularities on the hypersurface in which the coefficients have the conormal singularities.展开更多
This paper is concerned with the existence of nontrivial solutions of the somilinear olliptio equations on unbounded domain Ω in R^n, N≥2. These results extend Rose of [11], [13] and [16]. The main tool is variation...This paper is concerned with the existence of nontrivial solutions of the somilinear olliptio equations on unbounded domain Ω in R^n, N≥2. These results extend Rose of [11], [13] and [16]. The main tool is variational methods in this paper.展开更多
In this paper we study the blow-up behavior for a class of semilinear parabolic variational inequalities;whereK = {u ∈L<sup>2</sup>（0,T;H<sub>0</sub><sup>1</sup>（Ω））|u（x,t）...In this paper we study the blow-up behavior for a class of semilinear parabolic variational inequalities;whereK = {u ∈L<sup>2</sup>（0,T;H<sub>0</sub><sup>1</sup>（Ω））|u（x,t）≥ψ（x） a. e. （x,t） ∈Ω×（0,T）, u（x,0） = （x）},andis a uniformly elliptic operator.We prove the following main theorem.Theorem Let u（x,t） be a local solution of problem （I）,u∈C（0,T;H<sup>2</sup>（Ω）∩H<sub>0</sub><sup>1</sup>（Q））,u<sub>i</sub>∈L<sup>2</sup>（0,T;L<sup>2</sup>（Ω））, and following conditions are satisfied.（1） There exists a continuously differentiable function G（x,s） and a positive number α,such展开更多
文摘In this paper,we establish global classical solutions of semilinear wave equations with small compact supported initial data posed on the product space R^(3)×T.The semilinear nonlinearity is assumed to be of the cubic form.The main ingredient here is the establishment of the L^(2)-L^(∞)decay estimates and the energy estimates for the linear problem,which are adapted to the wave equation on the product space.The proof is based on the Fourier mode decomposition of the solution with respect to the periodic direction,the scaling technique,and the combination of the decay estimates and the energy estimates.
基金supported by the Zhejiang Provincial Natural Science Foundation of China(LY21A010016)the National Natural Science Foundation of China(11901550).
文摘In this paper, we investigate a blow-up phenomenon for a semilinear parabolic system on locally finite graphs. Under some appropriate assumptions on the curvature condition CDE’(n,0), the polynomial volume growth of degree m, the initial values, and the exponents in absorption terms, we prove that every non-negative solution of the semilinear parabolic system blows up in a finite time. Our current work extends the results achieved by Lin and Wu (Calc Var Partial Differ Equ, 2017, 56: Art 102) and Wu (Rev R Acad Cien Serie A Mat, 2021, 115: Art 133).
文摘In this paper, we present the a posteriori error estimate of two-grid mixed finite element methods by averaging techniques for semilinear elliptic equations. We first propose the two-grid algorithms to linearize the mixed method equations. Then, the averaging technique is used to construct the a posteriori error estimates of the two-grid mixed finite element method and theoretical analysis are given for the error estimators. Finally, we give some numerical examples to verify the reliability and efficiency of the a posteriori error estimator.
基金The project supported by NNSF of China(10071080)
文摘In this paper an existence and uniqueness theorem of positive solutions to a class of semilinear elliptic systems is proved. Also, a necessary condition for the existence of the positive solution is obtained. As the application of the main theorem, two examples are given.
基金the post-doctoral funds of China and funds of State Educational Commission of China for returned scholars from abroad
文摘An optimization theoretic approach of coefficients in semilinear parabolic equation is presented. It is based on convex analysis techniques. General theorems on existence are proved in L1 setting. A necessary condition is given for the solutions of the parameter estimatioll problem.
基金The research was supported in part by the grant of ZARCF and NSFC
文摘This paper considers the initial and boundary value problem of some linear and semilinear Schrodinger equation with real potential and H01 initial data. The author obtains the homogenization of linear and semilinear Schrodinger equations and gives correctors for the homogenization of linear and semilinear Schrodinger equations.
文摘An existence theorem is obtained for nonzero W^1,2(R^N) solutions of the following equations on R^N.-△u+ b(x)u = f(x,u), x∈R^N,where b is periodic for some variables and coercive for the others, f is superlinear.
文摘This paper is concerned with the existence and the nodal character of the nontrivial solutions in the entire space for some semilinear elliptic equation involving critical Sobolev exponent.
基金Supported by the Natural Science Foundation of China(10901126)
文摘The existence and uniqueness of singular solutions decaying like r m(see(1.4)) of the equation u + k i=1 ci|x|li upi = 0,x ∈ Rn(0.1) are obtained,where n ≥ 3,ci > 0,li > 2,i = 1,2,...,k,pi > 1,i = 1,2,...,k and the separation structure of singular solutions decaying like r(n 2) of eq.(0.1) are discussed.moreover,we obtain the explicit critical exponent ps(l)(see(1.9)).
基金TheNationalNaturalScienceFoundationofChina (No :10 0 710 4 8)
文摘The singularly perturbed boundary value problems for the semilinear elliptic equation are considered. Under suitable conditions and by using the fixed point theorem, the existence, uniqueness and asymptotic behavior of solution for the boundary value problems are studied.
基金the National Natural Science Foundation of ChinaFoundation for Fundamental Sciences of Nanchang UniversityHua-chen Found
文摘We discussed the Dirichlet problem of semilinear elliptic equation (Pβ,ε): -β2△u=up+εe,u>0, in Ω ;u=0, on Ω, where ΩcR"(N≤4) is smooth and bounded domain, p= β,ε>0. We have proved that there exist positive ε0. and ε1, such that when 0≤ε≤ε0, β>ε1, (Pβ,ε) has a single-peaked solution uβ,ε; furthermore, | uβ,ε|2-0 in the sense of measure as ε-0 and β-0.
基金supported by National Natural Science Foundation of China(11771284)
文摘In this paper, we study a semilinear Timoshenko system with heat conduction having two damping effects. The observation that two damping effects might lead to smaller decay rates for solutions in comparison to one damping effect is rigorously proved here in providing optimality results. Moreover, the global well-posedness for small data is presented.
文摘The numerical solution of a singularly perturbed problem for the semilinear parabolicdifferential equation with parabolic boundary layers is discussed.A nonlinear two-leveldifference scheme is constructed on the special non-uniform grids.The uniform convergenceof this scheme is proved and some numerical examples are given.
文摘This paper deals with the blow-up properties of solutions to semilinear heat equation ut-uxx= up in (0, 1) × (0, T) with the Neumann boundary condition ux(0, t) = 0, u:x1, t) = 1 on [0, T). The necessary and sufficient conditions under which all solutions to have a finite time blow-up and the exact blow-up rates are established. It is proved that the blow-up will occur only at the boundary x = 1. The asymptotic behavior near the blow-up time is also studied.
基金This work is supported by the National Science Foundation of China.
文摘In this paper, we will consider following initial value problem of semilinear stochastic evolution equation in Hilbert Space: where W（t） is a wiener process in H, H and Y are two real separable Hilbert Spaces, A is an infinitesimal generator of a strongly continuous semigroup s（t） on Y,f（t, y）: [0, T]×Y→Y, and G（t, y）: [0, T]×Y→L（H, Y）, y<sub>0</sub>: Ω→Y is a ramdom variable of square integrable. We apply theory of the semigroup and obtain two conclusions of uniqueness of the mild solution of （1） which include the corresponding results in [4].
文摘In the studies of nonlinear partial differential equations, the influence, from the singularities of coefficients to the singularities of solution, is a field that has not been dealt with. In this paper, we discuss a simple case of semilinear equations under the frame of the space of conormal distributions. We prove the result that the solution has the same singularities on the hypersurface in which the coefficients have the conormal singularities.
文摘This paper is concerned with the existence of nontrivial solutions of the somilinear olliptio equations on unbounded domain Ω in R^n, N≥2. These results extend Rose of [11], [13] and [16]. The main tool is variational methods in this paper.
文摘In this paper we study the blow-up behavior for a class of semilinear parabolic variational inequalities;whereK = {u ∈L<sup>2</sup>（0,T;H<sub>0</sub><sup>1</sup>（Ω））|u（x,t）≥ψ（x） a. e. （x,t） ∈Ω×（0,T）, u（x,0） = （x）},andis a uniformly elliptic operator.We prove the following main theorem.Theorem Let u（x,t） be a local solution of problem （I）,u∈C（0,T;H<sup>2</sup>（Ω）∩H<sub>0</sub><sup>1</sup>（Q））,u<sub>i</sub>∈L<sup>2</sup>（0,T;L<sup>2</sup>（Ω））, and following conditions are satisfied.（1） There exists a continuously differentiable function G（x,s） and a positive number α,such
