For a class of mixed initial-boundary value problem for general quasilinear hyperbolic systems, this paper establishes the local exact boundary controllability with boundary controls only acting on one end. As an appl...For a class of mixed initial-boundary value problem for general quasilinear hyperbolic systems, this paper establishes the local exact boundary controllability with boundary controls only acting on one end. As an application, the authors show the local exact boundary controllability for a kind of nonlinear vibrating string problem.展开更多
In this paper,we consider nonlinear degenerate quasilinear parabolic initial boundary value problems of second order.Using results from the theory of pseudomonotone operators,we show that there exists at least one wea...In this paper,we consider nonlinear degenerate quasilinear parabolic initial boundary value problems of second order.Using results from the theory of pseudomonotone operators,we show that there exists at least one weak solution in a suitable weighted Sobolev space.展开更多
In this paper,the study of gradient regularity for solutions of a class of elliptic problems of p-Laplace type is offered.In particular,we prove a global result concerning Lorentz-Morrey regularity of the non-homogene...In this paper,the study of gradient regularity for solutions of a class of elliptic problems of p-Laplace type is offered.In particular,we prove a global result concerning Lorentz-Morrey regularity of the non-homogeneous boundary data problem:-div((s^(2)+|▽u|^(2)p-2/2)▽u)=-div(|f|^(p-2)f)+g inΩ,u=h in■Ω,with the(sub-elliptic)degeneracy condition s∈[0,1]and with mixed data f∈L^(p)(Q;R^(n)),g∈Lp/(p-1)(Ω;R^(n))for p∈(1,n).This problem naturally arises in various applications such as dynamics of non-Newtonian fluid theory,electro-rheology,radiation of heat,plastic moulding and many others.Building on the idea of level-set inequality on fractional maximal distribution functions,it enables us to carry out a global regularity result of the solution via fractional maximal operators.Due to the significance of M_(α)and its relation with Riesz potential,estimates via fractional maximal functions allow us to bound oscillations not only for solution but also its fractional derivatives of orderα.Our approach therefore has its own interest.展开更多
Alinhac solved a long-standing open problem in 2001 and established that quasilinear wave equations in two space dimensions with quadratic null nonlinearities admit global-in-time solutions,provided that the initial d...Alinhac solved a long-standing open problem in 2001 and established that quasilinear wave equations in two space dimensions with quadratic null nonlinearities admit global-in-time solutions,provided that the initial data are compactly supported and sufficiently small in Sobolev norm.In this work,Alinhac obtained an upper bound with polynomial growth in time for the top-order energy of the solutions.A natural question then arises whether the time-growth is a true phenomenon,despite the possible conservation of basic energy.In the present paper,we establish that the top-order energy of the solutions in Alinhac theorem remains globally bounded in time.展开更多
In the paper, existence results for degenerate parabolic boundary value problems of higher order are proved. The weak solution is sought in a suitable weighted Sobolev space by using the generalized degree theory.
We study the following quasilinear Schrodinger equation-△u+V(x)u-△(u^(2))u=K(x)g(u),x∈R^(3),where the nonlinearity g(u)is asymptotically cubic at infinity,the potential V(x)may vanish at infinity.Under appropriate ...We study the following quasilinear Schrodinger equation-△u+V(x)u-△(u^(2))u=K(x)g(u),x∈R^(3),where the nonlinearity g(u)is asymptotically cubic at infinity,the potential V(x)may vanish at infinity.Under appropriate assumptions on K(x),we establish the existence of a nontrivial solution by using the mountain pass theorem.展开更多
We consider the following quasilinear Schrodinger equation involving p-Laplacian-Δpu+V(x)|u|^(p-2)u-Δp(|u|^(2η))|u|^(2η-2)u=λ|u|^(q-2)u/|x|^(μ)+|u|^(2ηp*(v)-2)u/|x|^(v)in R^(N),where N>p>1,η≥p/2(p-1),p&...We consider the following quasilinear Schrodinger equation involving p-Laplacian-Δpu+V(x)|u|^(p-2)u-Δp(|u|^(2η))|u|^(2η-2)u=λ|u|^(q-2)u/|x|^(μ)+|u|^(2ηp*(v)-2)u/|x|^(v)in R^(N),where N>p>1,η≥p/2(p-1),p<q<2ηp^(*)(μ),p^(*)(s)=(p(N-s))/N-p,andλ,μ,νare parameters withλ>0,μ,ν∈[0,p).Via the Mountain Pass Theorem and the Concentration Compactness Principle,we establish the existence of nontrivial ground state solutions for the above problem.展开更多
The authors study the existence of standing wave solutions for the quasilinear Schr?dinger equation with the critical exponent and singular coefficients.By applying the mountain pass theorem and the concentration comp...The authors study the existence of standing wave solutions for the quasilinear Schr?dinger equation with the critical exponent and singular coefficients.By applying the mountain pass theorem and the concentration compactness principle,they get a ground state solution.Moreover,the asymptotic behavior of the ground state solution is also obtained.展开更多
In this paper, we get the existence of a weak solution of the following inhomogeneous quasilinear elliptic equation with critical growth conditions: where N≥2, f（x,u）～|u|<sup>m-1</sup>e<sup>b|u|&...In this paper, we get the existence of a weak solution of the following inhomogeneous quasilinear elliptic equation with critical growth conditions: where N≥2, f（x,u）～|u|<sup>m-1</sup>e<sup>b|u|<sup>γ</sup></sup>at +∞, with γ=N/N-1, m≥1, b】0.展开更多
We consider a class of modified quasilinear Schrodinger equations-△u+k/2u△u^(2)=λα(x)u^(-α)+b(x)u^(β) in Ω with u(x)=0 on■Ω,where Ω■R^(N)is a bounded domain with a regular boundary,N≥3,a and b are bounded ...We consider a class of modified quasilinear Schrodinger equations-△u+k/2u△u^(2)=λα(x)u^(-α)+b(x)u^(β) in Ω with u(x)=0 on■Ω,where Ω■R^(N)is a bounded domain with a regular boundary,N≥3,a and b are bounded mensurable functions,0<α<1<β<2*-1 and k,λ≥0 are two parameters.We establish the global existence and multiplicity results of positive solutions in H^(1)_(0)(Ω)∩L^(∞)(Ω)for appropriate classes of parameters k andλand coefficients a(x)and b(x).展开更多
We consider the Poiseuille flow of nematic liquid crystals via the full Ericksen-Leslie model.The model is described by a coupled system consisting of a heat equation and a quasilinear wave equation.In this paper,we w...We consider the Poiseuille flow of nematic liquid crystals via the full Ericksen-Leslie model.The model is described by a coupled system consisting of a heat equation and a quasilinear wave equation.In this paper,we will construct an example with a finite time cusp singularity due to the quasilinearity of the wave equation,extended from an earlier resultonaspecial case.展开更多
基金Project supported by the Special Funds forMajor State Basic Research Projects ofChina.
文摘For a class of mixed initial-boundary value problem for general quasilinear hyperbolic systems, this paper establishes the local exact boundary controllability with boundary controls only acting on one end. As an application, the authors show the local exact boundary controllability for a kind of nonlinear vibrating string problem.
基金This research is supported by the Natural science Foundation of Hunan province
文摘In this paper,we consider nonlinear degenerate quasilinear parabolic initial boundary value problems of second order.Using results from the theory of pseudomonotone operators,we show that there exists at least one weak solution in a suitable weighted Sobolev space.
基金supported by Ministry of Education and Training(Vietnam),under grant number B2023-SPS-01。
文摘In this paper,the study of gradient regularity for solutions of a class of elliptic problems of p-Laplace type is offered.In particular,we prove a global result concerning Lorentz-Morrey regularity of the non-homogeneous boundary data problem:-div((s^(2)+|▽u|^(2)p-2/2)▽u)=-div(|f|^(p-2)f)+g inΩ,u=h in■Ω,with the(sub-elliptic)degeneracy condition s∈[0,1]and with mixed data f∈L^(p)(Q;R^(n)),g∈Lp/(p-1)(Ω;R^(n))for p∈(1,n).This problem naturally arises in various applications such as dynamics of non-Newtonian fluid theory,electro-rheology,radiation of heat,plastic moulding and many others.Building on the idea of level-set inequality on fractional maximal distribution functions,it enables us to carry out a global regularity result of the solution via fractional maximal operators.Due to the significance of M_(α)and its relation with Riesz potential,estimates via fractional maximal functions allow us to bound oscillations not only for solution but also its fractional derivatives of orderα.Our approach therefore has its own interest.
基金supported by the China Postdoctoral Science Foundation(2021M690702)The author Z.L.was in part supported by NSFC(11725102)+2 种基金Sino-German Center(M-0548)the National Key R&D Program of China(2018AAA0100303)National Support Program for Young Top-Notch TalentsShanghai Science and Technology Program[21JC1400600 and No.19JC1420101].
文摘Alinhac solved a long-standing open problem in 2001 and established that quasilinear wave equations in two space dimensions with quadratic null nonlinearities admit global-in-time solutions,provided that the initial data are compactly supported and sufficiently small in Sobolev norm.In this work,Alinhac obtained an upper bound with polynomial growth in time for the top-order energy of the solutions.A natural question then arises whether the time-growth is a true phenomenon,despite the possible conservation of basic energy.In the present paper,we establish that the top-order energy of the solutions in Alinhac theorem remains globally bounded in time.
基金Supported by the funds of the State Educational Commission of China for returned scholars from abroad
文摘In the paper, existence results for degenerate parabolic boundary value problems of higher order are proved. The weak solution is sought in a suitable weighted Sobolev space by using the generalized degree theory.
基金the National Natural Science Foundation of China(No.11901499 and No.11901500)Nanhu Scholar Program for Young Scholars of XYNU(No.201912)。
文摘We study the following quasilinear Schrodinger equation-△u+V(x)u-△(u^(2))u=K(x)g(u),x∈R^(3),where the nonlinearity g(u)is asymptotically cubic at infinity,the potential V(x)may vanish at infinity.Under appropriate assumptions on K(x),we establish the existence of a nontrivial solution by using the mountain pass theorem.
基金supported by the National Natural Science Foundation of China (12226411)the Research Ability Cultivation Fund of HUAS (No.2020kypytd006)+1 种基金supported by the National Natural Science Foundation of China (11931012,11871386)the Fundamental Research Funds for the Central Universities (WUT:2020IB019)。
文摘We consider the following quasilinear Schrodinger equation involving p-Laplacian-Δpu+V(x)|u|^(p-2)u-Δp(|u|^(2η))|u|^(2η-2)u=λ|u|^(q-2)u/|x|^(μ)+|u|^(2ηp*(v)-2)u/|x|^(v)in R^(N),where N>p>1,η≥p/2(p-1),p<q<2ηp^(*)(μ),p^(*)(s)=(p(N-s))/N-p,andλ,μ,νare parameters withλ>0,μ,ν∈[0,p).Via the Mountain Pass Theorem and the Concentration Compactness Principle,we establish the existence of nontrivial ground state solutions for the above problem.
基金supported by the National Natural Science Foundation of China(Nos.11971393,11901499,11801465)Nanhu Scholar Program for Young Scholars of XYNU(No.201912)。
文摘The authors study the existence of standing wave solutions for the quasilinear Schr?dinger equation with the critical exponent and singular coefficients.By applying the mountain pass theorem and the concentration compactness principle,they get a ground state solution.Moreover,the asymptotic behavior of the ground state solution is also obtained.
基金Supported by the Youth FoundationNatural Science Foundation, People's Republic of China.
文摘In this paper, we get the existence of a weak solution of the following inhomogeneous quasilinear elliptic equation with critical growth conditions: where N≥2, f（x,u）～|u|<sup>m-1</sup>e<sup>b|u|<sup>γ</sup></sup>at +∞, with γ=N/N-1, m≥1, b】0.
基金supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq/Brazil) (Grant No.311562/2020-5)supported by National Natural Science Foundation of China (Grant Nos.11971436 and 12011530199)+1 种基金Natural Science Foundation of Zhejiang (Grant Nos.LZ22A010001 and LD19A010001)supported by Coordenacao de Aperfei coamento de Pessoal de Nível Superior (CAPES/Brazil) (Grant No.2788/2015-02)。
文摘We consider a class of modified quasilinear Schrodinger equations-△u+k/2u△u^(2)=λα(x)u^(-α)+b(x)u^(β) in Ω with u(x)=0 on■Ω,where Ω■R^(N)is a bounded domain with a regular boundary,N≥3,a and b are bounded mensurable functions,0<α<1<β<2*-1 and k,λ≥0 are two parameters.We establish the global existence and multiplicity results of positive solutions in H^(1)_(0)(Ω)∩L^(∞)(Ω)for appropriate classes of parameters k andλand coefficients a(x)and b(x).
文摘We consider the Poiseuille flow of nematic liquid crystals via the full Ericksen-Leslie model.The model is described by a coupled system consisting of a heat equation and a quasilinear wave equation.In this paper,we will construct an example with a finite time cusp singularity due to the quasilinearity of the wave equation,extended from an earlier resultonaspecial case.
