In this paper, the existence and the uniqueness of the local generalized solution and the local classical solution of the Cauchy problem for the generalized BBM-Burgers equationare proved. The existence and the unique...In this paper, the existence and the uniqueness of the local generalized solution and the local classical solution of the Cauchy problem for the generalized BBM-Burgers equationare proved. The existence and the uniqueness of the global generalized solution and the global classical solution for the Cauchy problem of equation (1) are proved when n = 3, 2, 1. Moreover, the decay property of the solution is discussed.展开更多
This article considers Cauchy problem u(t) - (uv)(x) = 0, v(t) - u(x) = 0, u(x, 0) = u(0) (x) > 0, v(x, 0) = v(0)(x). A necessary and sufficient condition in guaranteeing that Cauchy problem admits a global C-1-sol...This article considers Cauchy problem u(t) - (uv)(x) = 0, v(t) - u(x) = 0, u(x, 0) = u(0) (x) > 0, v(x, 0) = v(0)(x). A necessary and sufficient condition in guaranteeing that Cauchy problem admits a global C-1-solution on t greater than or equal to 0 is obtained.展开更多
This article considers Cauchy problem for quasilinear hyperbolic systems in diagonal form. A necessary and sufficient condition in guaranteeing that Cauchy problem admits a unique global classical solution on t ≥ 0 i...This article considers Cauchy problem for quasilinear hyperbolic systems in diagonal form. A necessary and sufficient condition in guaranteeing that Cauchy problem admits a unique global classical solution on t ≥ 0 is obtained, and a sharp estimate of the life span for the classical solution is given.展开更多
The main goal of the paper is to obtain the local strong solution of the Cauchy problem of the nonhomogeneous incompressible Boussinesq equation in two-dimension space. Especially, when the far-field density is vacuum...The main goal of the paper is to obtain the local strong solution of the Cauchy problem of the nonhomogeneous incompressible Boussinesq equation in two-dimension space. Especially, when the far-field density is vacuum, we make a priori estimate in a bound ball and prove the existence and uniqueness of the local strong solution of the Boussinesq equation.展开更多
For a continuous, increasing function ω : R^+ →R^+/{0} of finite exponential type, this paper introduces the set Z(A, ω) of all x in a Banach space X for which the second order abstract differential equation ...For a continuous, increasing function ω : R^+ →R^+/{0} of finite exponential type, this paper introduces the set Z(A, ω) of all x in a Banach space X for which the second order abstract differential equation (2) has a mild solution such that [ω(t)]^-1u(t,x) is uniformly continues on R^+, and show that Z(A, ω) is a maximal Banach subspace continuously embedded in X, where A ∈ B(X) is closed. Moreover, A[z(A,ω) generates an O(ω(t)) strongly continuous cosine operator function family.展开更多
Integrated semigroups which are not exponentially bounded have been studied. The integrated solutions of inhomogeneous and nonlinear abstract Caucby problems are discussed. The reults extends and improves the theorems...Integrated semigroups which are not exponentially bounded have been studied. The integrated solutions of inhomogeneous and nonlinear abstract Caucby problems are discussed. The reults extends and improves the theorems in [1].展开更多
We prove existence and uniqueness of the global solution to the Cauchy problem on a universe fireworks model with finite total mass at the initial state when the ratio of the mass surviving the explosion, the probabil...We prove existence and uniqueness of the global solution to the Cauchy problem on a universe fireworks model with finite total mass at the initial state when the ratio of the mass surviving the explosion, the probability of the explosion of fragments and the probability function of the velocity change of a surviving particle satisfy the corresponding physical conditions. Although the nonrelativistic Boltzmann-like equation modeling the universe fireworks is mathematically easy, this article leads rather theoretically to an understanding of how to construct contractive mappings in a Banach space for the proof of the existence and uniqueness of the solution by means of methods taken from the famous work by DiPerna & Lions about the Boltzmann equation. We also show both the regularity and the time-asymptotic behavior of solution to the Cauchy problem.展开更多
Concerns with the nonexistence of global solutions to the initial boundary value problem for a nonlinear hyperbolic equation with material damping. Nonexitence theorems of global solutions to the above problem are pro...Concerns with the nonexistence of global solutions to the initial boundary value problem for a nonlinear hyperbolic equation with material damping. Nonexitence theorems of global solutions to the above problem are proved by the energy method, Jensen inequality and the concavity method, respectively. As applications of our main results, three examples are given.展开更多
In this paper we study the decay estimate of global solutions to the initial-boundary value problem for double degenerate nonlinear parabolic equation by using a dif-ference inequality.
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 article, we prove that the Cauchy problem for a N-dimensional system of nonlinear wave equations…… admits a unique global generalized solution in ……and a unique global classical solution in…… the suffici...In this article, we prove that the Cauchy problem for a N-dimensional system of nonlinear wave equations…… admits a unique global generalized solution in ……and a unique global classical solution in…… the sufficient conditions of the blow up of the solution in finite time are given, and also two examples are given.展开更多
In the present paper,the local existence of classical solutions to the periodic boundary problem and the Cauchy problem of a quasilinear evolution equation are studied under the assumptions that do not require the mon...In the present paper,the local existence of classical solutions to the periodic boundary problem and the Cauchy problem of a quasilinear evolution equation are studied under the assumptions that do not require the monotonicity of σi(s) (i= 1,…, n). The nonexistence of global solutions to the initial-boundary value problem of the equation is also discussed, a blowup theorem is proved and a concrete example is given.展开更多
The paper concerns with the existence, uniqueness and nonexistence of global solution to the Cauchy problem for a class of nonlinear wave equations with damping term. It proves that under suitable assumptions on nonli...The paper concerns with the existence, uniqueness and nonexistence of global solution to the Cauchy problem for a class of nonlinear wave equations with damping term. It proves that under suitable assumptions on nonlinear the function and initial data the abovementioned problem admits a unique global solution by Fourier transform method. The sufficient conditions of nonexistence of the global solution to the above-mentioned problem are given by the concavity method.展开更多
In this paper we study solvability of the Cauchy problem of the Kawahara equation 偏导dtu + au偏导dzu + β偏导d^3xu +γ偏导d^5xu = 0 with L^2 initial data. By working on the Bourgain space X^r,s(R^2) associated w...In this paper we study solvability of the Cauchy problem of the Kawahara equation 偏导dtu + au偏导dzu + β偏导d^3xu +γ偏导d^5xu = 0 with L^2 initial data. By working on the Bourgain space X^r,s(R^2) associated with this equation, we prove that the Cauchy problem of the Kawahara equation is locally solvable if initial data belong to H^r(R) and -1 〈 r ≤ 0. This result combined with the energy conservation law of the Kawahara equation yields that global solutions exist if initial data belong to L^2(R).展开更多
In this paper, the existence and uniqueness of the global generalized solution and the global classical solution for the Cauchy problem of the singularly perturbed sixth order Boussinesq type equation are proved.
We consider the singularly perturbed sixth-order Boussinesq-type equation, which describes the bidirectional propagation of small amplitude and long capillary gravity waves on the surface of shallow water for bond num...We consider the singularly perturbed sixth-order Boussinesq-type equation, which describes the bidirectional propagation of small amplitude and long capillary gravity waves on the surface of shallow water for bond number (surface tension parameter) less than but very close to 1/3. The sufficient conditions of blow-up of solution to the Cauchy problem for this equation are given.展开更多
The infinite generator of α-times Integrated C semigroups and the properties of resolvent are given. At the same time, we discuss the relationship between the existence of strong solution of a class of nonhomogeneous...The infinite generator of α-times Integrated C semigroups and the properties of resolvent are given. At the same time, we discuss the relationship between the existence of strong solution of a class of nonhomogeneous abstract Cauchy problem and α-times integrated C semigroups, and a sufficient and necessary condition is obtained.展开更多
In this paper we study the asymptotic behavior of global classical solutions to the Cauchy problem with initial data given on a semi-bounded axis for quasilinear hyperbolic systems. Based on the existence result on th...In this paper we study the asymptotic behavior of global classical solutions to the Cauchy problem with initial data given on a semi-bounded axis for quasilinear hyperbolic systems. Based on the existence result on the global classical solution, we prove that, when t tends to the infinity, the solution approaches a combination of C1 travelling wave solutions with the algebraic rate (1 + t)^-u, provided that the initial data decay with the rate (1 + x)^-(l+u) (resp. (1 - x)^-(1+u)) as x tends to +∞ (resp. -∞), where u is a positive constant.展开更多
The authors consider the Cauchy problem with a kind of non-smooth initial data for quasilinear hyperbolic systems and obtain a necessary and sufficient condition to guarantee the existence and uniqueness of global wea...The authors consider the Cauchy problem with a kind of non-smooth initial data for quasilinear hyperbolic systems and obtain a necessary and sufficient condition to guarantee the existence and uniqueness of global weakly discontinuous solution.展开更多
基金supported by the National Natural Science Foundation of China(11226175,11271336 and 11171311)Specialized Reseach Fund for the Docotoral Program of Higher Education(20124301120002)Foundation of He’nan Educational Committee(2009C110006)
文摘In this paper, the existence and the uniqueness of the local generalized solution and the local classical solution of the Cauchy problem for the generalized BBM-Burgers equationare proved. The existence and the uniqueness of the global generalized solution and the global classical solution for the Cauchy problem of equation (1) are proved when n = 3, 2, 1. Moreover, the decay property of the solution is discussed.
基金Project supported by the NSF of Fujian Province (A97020)
文摘This article considers Cauchy problem u(t) - (uv)(x) = 0, v(t) - u(x) = 0, u(x, 0) = u(0) (x) > 0, v(x, 0) = v(0)(x). A necessary and sufficient condition in guaranteeing that Cauchy problem admits a global C-1-solution on t greater than or equal to 0 is obtained.
基金Project supported by the NSF of China! (19971O62)the NSF of Fujian Province!(A97020) the NSF of Educational Committee of
文摘This article considers Cauchy problem for quasilinear hyperbolic systems in diagonal form. A necessary and sufficient condition in guaranteeing that Cauchy problem admits a unique global classical solution on t ≥ 0 is obtained, and a sharp estimate of the life span for the classical solution is given.
文摘The main goal of the paper is to obtain the local strong solution of the Cauchy problem of the nonhomogeneous incompressible Boussinesq equation in two-dimension space. Especially, when the far-field density is vacuum, we make a priori estimate in a bound ball and prove the existence and uniqueness of the local strong solution of the Boussinesq equation.
文摘For a continuous, increasing function ω : R^+ →R^+/{0} of finite exponential type, this paper introduces the set Z(A, ω) of all x in a Banach space X for which the second order abstract differential equation (2) has a mild solution such that [ω(t)]^-1u(t,x) is uniformly continues on R^+, and show that Z(A, ω) is a maximal Banach subspace continuously embedded in X, where A ∈ B(X) is closed. Moreover, A[z(A,ω) generates an O(ω(t)) strongly continuous cosine operator function family.
文摘Integrated semigroups which are not exponentially bounded have been studied. The integrated solutions of inhomogeneous and nonlinear abstract Caucby problems are discussed. The reults extends and improves the theorems in [1].
基金supported by NSFC(10271121)the Scientific Research Foundation for the Returned Overseas Chinese Scholars,the Education Ministry of China+5 种基金sponsored by joint grants of NSFC 10511120278/10611120371RFBR 04-02-39026supported by NSFC (10671130)E-Institutes of Shanghai Municipal Education Commission (E03004)Shanghai Science and Technology Commission (06JC14092)Shuguang Project of Shanghai Municipal Education Commission (06SG45)
文摘We prove existence and uniqueness of the global solution to the Cauchy problem on a universe fireworks model with finite total mass at the initial state when the ratio of the mass surviving the explosion, the probability of the explosion of fragments and the probability function of the velocity change of a surviving particle satisfy the corresponding physical conditions. Although the nonrelativistic Boltzmann-like equation modeling the universe fireworks is mathematically easy, this article leads rather theoretically to an understanding of how to construct contractive mappings in a Banach space for the proof of the existence and uniqueness of the solution by means of methods taken from the famous work by DiPerna & Lions about the Boltzmann equation. We also show both the regularity and the time-asymptotic behavior of solution to the Cauchy problem.
基金Project supported by the National Natural Science Foundation of China (Nos. 10371073 and 10572156) the Natural Science Foundation of Henan Province of China (No.0611050500)
文摘Concerns with the nonexistence of global solutions to the initial boundary value problem for a nonlinear hyperbolic equation with material damping. Nonexitence theorems of global solutions to the above problem are proved by the energy method, Jensen inequality and the concavity method, respectively. As applications of our main results, three examples are given.
基金Supported by the NNSF of China(10441002)Supported by NNSF of Henan Province(200510466011)
文摘In this paper we study the decay estimate of global solutions to the initial-boundary value problem for double degenerate nonlinear parabolic equation by using a dif-ference inequality.
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.
基金supported by Tianyuan Youth Foundation of Mathematics (11226177)the National Natural Science Foundation of China (11271336 and 11171311)Foundation of He’nan Educational Committee (2009C110006)
文摘In this article, we prove that the Cauchy problem for a N-dimensional system of nonlinear wave equations…… admits a unique global generalized solution in ……and a unique global classical solution in…… the sufficient conditions of the blow up of the solution in finite time are given, and also two examples are given.
基金Natural Science Foundation of Henan Province!(Grant No.98405070) National Natural Science Foundation of China (Grant No.19
文摘In the present paper,the local existence of classical solutions to the periodic boundary problem and the Cauchy problem of a quasilinear evolution equation are studied under the assumptions that do not require the monotonicity of σi(s) (i= 1,…, n). The nonexistence of global solutions to the initial-boundary value problem of the equation is also discussed, a blowup theorem is proved and a concrete example is given.
基金Supported by the National Natural Science Foundation of China(10371073)
文摘The paper concerns with the existence, uniqueness and nonexistence of global solution to the Cauchy problem for a class of nonlinear wave equations with damping term. It proves that under suitable assumptions on nonlinear the function and initial data the abovementioned problem admits a unique global solution by Fourier transform method. The sufficient conditions of nonexistence of the global solution to the above-mentioned problem are given by the concavity method.
基金Project supported by the China National Natural Science Foundation (Grants 10171111, 10171112)
文摘In this paper we study solvability of the Cauchy problem of the Kawahara equation 偏导dtu + au偏导dzu + β偏导d^3xu +γ偏导d^5xu = 0 with L^2 initial data. By working on the Bourgain space X^r,s(R^2) associated with this equation, we prove that the Cauchy problem of the Kawahara equation is locally solvable if initial data belong to H^r(R) and -1 〈 r ≤ 0. This result combined with the energy conservation law of the Kawahara equation yields that global solutions exist if initial data belong to L^2(R).
文摘In this paper, the existence and uniqueness of the global generalized solution and the global classical solution for the Cauchy problem of the singularly perturbed sixth order Boussinesq type equation are proved.
文摘We consider the singularly perturbed sixth-order Boussinesq-type equation, which describes the bidirectional propagation of small amplitude and long capillary gravity waves on the surface of shallow water for bond number (surface tension parameter) less than but very close to 1/3. The sufficient conditions of blow-up of solution to the Cauchy problem for this equation are given.
文摘The infinite generator of α-times Integrated C semigroups and the properties of resolvent are given. At the same time, we discuss the relationship between the existence of strong solution of a class of nonhomogeneous abstract Cauchy problem and α-times integrated C semigroups, and a sufficient and necessary condition is obtained.
基金Supported by the National Natural Science Foundation of China (Grant No.10771038)
文摘In this paper we study the asymptotic behavior of global classical solutions to the Cauchy problem with initial data given on a semi-bounded axis for quasilinear hyperbolic systems. Based on the existence result on the global classical solution, we prove that, when t tends to the infinity, the solution approaches a combination of C1 travelling wave solutions with the algebraic rate (1 + t)^-u, provided that the initial data decay with the rate (1 + x)^-(l+u) (resp. (1 - x)^-(1+u)) as x tends to +∞ (resp. -∞), where u is a positive constant.
基金Project supported by the Special Funds for Major State Basic Research Projects of China
文摘The authors consider the Cauchy problem with a kind of non-smooth initial data for quasilinear hyperbolic systems and obtain a necessary and sufficient condition to guarantee the existence and uniqueness of global weakly discontinuous solution.