This paper is concerned with radially positive solutions of the k-Hessian equation involving a Matukuma-type source S_(k)(D^(2)(-φ))=|x|^(λ-2)/(1+|x|^(2))^(λ2)φ^(q),x∈Ω,where S_(k)(D^(2)(-φ))is the k-Hessian op...This paper is concerned with radially positive solutions of the k-Hessian equation involving a Matukuma-type source S_(k)(D^(2)(-φ))=|x|^(λ-2)/(1+|x|^(2))^(λ2)φ^(q),x∈Ω,where S_(k)(D^(2)(-φ))is the k-Hessian operator,q>k>1,λ>0,n>2k,k∈N,andΩis a suitable bounded do-main in R~n.It turns out that there are two different types of radially positive solutions for k>1,i.e.,M-solution(singular at r=0)and Esolution(regular at r=0),which is distinct from the case when k=1.For 1<q<[(n-2+λ)k](n-2k),we apply an iterative approach to improve accuracy of asymptotic expansions of M-solution step by step to the desired extend.In contrast to the case k=1,we require a more precise range of parameters due to repeated application of Taylor expansions,which also makes asymptotic expansions need more delicate investigation.展开更多
This paper investigates the self-similar singular solution of the p-Laplacian evolution equation with the nonlinear gradient absorption terms u t=div(u p-2u)-uq for 1<p<2 and q>in Rn× (0, ∞). It has ...This paper investigates the self-similar singular solution of the p-Laplacian evolution equation with the nonlinear gradient absorption terms u t=div(u p-2u)-uq for 1<p<2 and q>in Rn× (0, ∞). It has been proved that when 1<q<p-n/(n+1) there exists a unique self-similar very singular solution.展开更多
In the present paper, closed form singular solutions for an infinite anisotropic plate with an elliptic hole or crack are derived based on the Stroh-type formalism for the general anisotropic plate. With the solutions...In the present paper, closed form singular solutions for an infinite anisotropic plate with an elliptic hole or crack are derived based on the Stroh-type formalism for the general anisotropic plate. With the solutions, the hoop stresses and hoop moments around the elliptic hole as well as the stress intensity factors at the crack tip under concentrated in-plane stresses and bending moments are obtained. The singular solutions can be used for approximate analysis of an anisotropic plate weakened by a hole or a crack under concentrated forces and moments. They can also be used as fundamental solutions of boundary integral equations in BEM analysis for anisotropic plates with holes or cracks under general force and boundary conditions.展开更多
This paper studies several problems , which are potentially relevant for the construction of adaptive numerical schemes. First, biorthogonal spline wavelets on [0,1] are chosen as a starting point for characterization...This paper studies several problems , which are potentially relevant for the construction of adaptive numerical schemes. First, biorthogonal spline wavelets on [0,1] are chosen as a starting point for characterizations of functions in Besom spaces B(?)(0,1) with 0<σ<∞ and (1+σ)-1<γ<∞. Such function spaces are known to be related to nonlinear approximation. Then so called restricted nonlinear approximation procedures with respect to Sobolev space norms are considered. Besides characterization results Jackson type estimates for various tree-type and tresholding algorithms are investigated. Finally known approximation results for geometry induced singularity functions of boundary integeral equations are combined with the characterization results for restricted nonlinear approximation to show Besov space regularity results.展开更多
The self-similar singular solution of the fast diffusion equation with nonlinear gradient absorption terms are studied. By a self-similar transformation, the self-similar solutions satisfy a boundary value problem of ...The self-similar singular solution of the fast diffusion equation with nonlinear gradient absorption terms are studied. By a self-similar transformation, the self-similar solutions satisfy a boundary value problem of nonlinear ordinary differential equation （ODE）. Using the shooting arguments, the existence and uniqueness of the solution to the initial data problem of the nonlinear ODE are investigated, and the solutions are classified by the region of the initial data. The necessary and sufficient condition for the existence and uniqueness of self-similar very singular solutions is obtained by investigation of the classification of the solutions. In case of existence, the self-similar singular solution is very singular solution.展开更多
The two-component Camassa–Holm equation includes many intriguing phenomena. We propose a multi-symplectic compact method to solve the two-component Camassa–Holm equation. Based on its multi-symplectic formulation, t...The two-component Camassa–Holm equation includes many intriguing phenomena. We propose a multi-symplectic compact method to solve the two-component Camassa–Holm equation. Based on its multi-symplectic formulation, the proposed method is derived by the sixth-order compact finite difference method in spatial discretization and the symplectic implicit midpoint scheme in temporal discretization. Numerical experiments finely describe the velocity and density variables in the two-component integrable system and distinctly display the evolvement of the singular solutions. Moreover, the proposed method shows good conservative properties during long-time numerical simulation.展开更多
This paper studies the Cauchy problem of the dissipative quasi-geostrophic equation in pseudomeasure space PM^n＋1-2α（R^n） or Lorentz space L n/2α-1,∞（R^n）, which admit the singular solutions. The global well-p...This paper studies the Cauchy problem of the dissipative quasi-geostrophic equation in pseudomeasure space PM^n＋1-2α（R^n） or Lorentz space L n/2α-1,∞（R^n）, which admit the singular solutions. The global well-posedness is established provided initial data θ0（x） are small enough in these spaces. Moreover, the asymptotic stability of solutions in pseudomeasure space is proved. In particular, if the initial data are homo-geneous functions of degree 1 - 2α, the self-similar solutions are also obtained.展开更多
The existence and uniqueness of singular solutions decaying like r^-m（see （1.4）） of the equation △u＋k∑i=1ci｜x｜liupi=0,x∈R^N are obtained, wheren≥3, ci 〉0, li〉-2, i=1,2,..,k, pi〉 1, i=l,2,-..,kandthe sepa...The existence and uniqueness of singular solutions decaying like r^-m（see （1.4）） of the equation △u＋k∑i=1ci｜x｜liupi=0,x∈R^N are obtained, wheren≥3, ci 〉0, li〉-2, i=1,2,..,k, pi〉 1, i=l,2,-..,kandthe 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
We study overdetermined systems of first order partial differential equations with singular solutions.The main result gives a characterization of such systems and asserts that the singular solution is equal to the con...We study overdetermined systems of first order partial differential equations with singular solutions.The main result gives a characterization of such systems and asserts that the singular solution is equal to the contact singular set.展开更多
The case of a radial initial state for a family of hyperbolic systems of con- servation laws with several spatial dimensions is considered. It will be shown that the singularity at the origin introduces multiple solut...The case of a radial initial state for a family of hyperbolic systems of con- servation laws with several spatial dimensions is considered. It will be shown that the singularity at the origin introduces multiple solutions outside of the traditional admissible classes.展开更多
We revise one case of the M-solutions of B. Wang, et al., J. Diff. Eqs. 253 （2012）, pp. 3232-3265 and obtain more precise form of the singular term and the regular term.
In this paper, we investigate the existence of positive solutions for the singular fourth-order differential system <em>u</em><sup>(4)</sup> = <em><span style="white-space:nowrap;...In this paper, we investigate the existence of positive solutions for the singular fourth-order differential system <em>u</em><sup>(4)</sup> = <em><span style="white-space:nowrap;">φ</span>u</em> + <em>f </em>(<em>t</em>, <em>u</em>, <em>u</em>”, <em><span style="white-space:nowrap;">φ</span></em>), 0 < <em>t</em> < 1, -<em><span style="white-space:nowrap;">φ</span></em>” = <em>μg</em> (<em>t</em>, <em>u</em>, <em>u</em>”), 0 < <em>t</em> < 1, <em>u</em> (0) = <em>u</em> (1) = <em>u</em>”(0) = <em>u</em>”(1) = 0, <em><span style="white-space:nowrap;">φ</span> </em>(0) = <em><span style="white-space:nowrap;">φ</span> </em>(1) = 0;where <em>μ</em> > 0 is a constant, and the nonlinear terms<em> f</em>, <em>g</em> may be singular with respect to both the time and space variables. The results obtained herein generalize and improve some known results including singular and non-singular cases.展开更多
In this paper, the difficulties on calculation in solving singular integral equations are overcome when the restriction of curve of integration to be a closed contour is cancelled. When the curve is an open arc and th...In this paper, the difficulties on calculation in solving singular integral equations are overcome when the restriction of curve of integration to be a closed contour is cancelled. When the curve is an open arc and the solutions for singular integral equations possess singularities of higher order, the solution and the solvable condition for characteristic equations as well as the generalized Noether theorem for complete equations are given.展开更多
The basic sets of solutions in classH(orH*)for the characteristic equation and its adjoint equation with Hilbert kernel are given respectively.Thus the expressions of solutions and its solvable conditions are simplifi...The basic sets of solutions in classH(orH*)for the characteristic equation and its adjoint equation with Hilbert kernel are given respectively.Thus the expressions of solutions and its solvable conditions are simplified.On this basis the solutions and the solvable conditions in classH_(1)as well as the generalized Noether theorem for the complete equation are obtained.展开更多
We transform the singular integral equations with solutions simultaneously having singularities of higher order at infinite point and at several finite points on the real axis into ones along a closed contour with sol...We transform the singular integral equations with solutions simultaneously having singularities of higher order at infinite point and at several finite points on the real axis into ones along a closed contour with solutions having singularities of higher order, and for the former obtain the extended Neother theorem of complete equation as well as the solutions and the solvable conditions of characteristic equation from the latter. The conclusions drawn by this article contain special cases discussed before.展开更多
Under the appropriate hypotheses subject to the unknown function and the free term, by means of our Lemma, we prove the rationality of order at x = ∞ on two sides for the characteristic singular integral equations wi...Under the appropriate hypotheses subject to the unknown function and the free term, by means of our Lemma, we prove the rationality of order at x = ∞ on two sides for the characteristic singular integral equations with solutions having singularities of higher order on the real axis X. We transform the equations into solving equivalent Riemann boundary value problems with solutions having singularities of higher order and with additional conditions on X. The solutions and the solvable conditions for the former are obtained from the latter.展开更多
In this paper we consider the singularly perturbed boundary value problem for the fourth-order elliptic differential equation, establish the energy estimates of the solutionand its derivatives and construct the formal...In this paper we consider the singularly perturbed boundary value problem for the fourth-order elliptic differential equation, establish the energy estimates of the solutionand its derivatives and construct the formal asymptotic solution by Lyuternik- Vishik 's method. Finally, by means of the energy estimates we obtain the bound of the remainder of the asymptotic solution.展开更多
We establish the existence of positive periodic solutions of the second-order singular coupled systems{x′′＋ p_1（t）x′＋ q_1（t）x = f_1（t, y（t）） ＋ c_1（t）,y′′＋ p_2（t）y′＋ q_2（t）y = f_2（t, x（t）） ...We establish the existence of positive periodic solutions of the second-order singular coupled systems{x′′＋ p_1（t）x′＋ q_1（t）x = f_1（t, y（t）） ＋ c_1（t）,y′′＋ p_2（t）y′＋ q_2（t）y = f_2（t, x（t）） ＋ c_2（t）,where pi, qi, ci ∈ C（R/T Z; R）, i = 1, 2; f_1, f_2 ∈ C（R/T Z ×（0, ∞）, R） and may be singular near the zero. The proof relies on Schauder＇s fixed point theorem and anti-maximum principle.Our main results generalize and improve those available in the literature.展开更多
基金Supported by the National Natural Science Foundation of China (11801436)the Research startup Foundation for Talent Introduction of Xi'an University of Science and Technology (2050123041)the Natural Science Basic Research Program of Shaanxi Province (2024JC-YBQN-0014)。
文摘This paper is concerned with radially positive solutions of the k-Hessian equation involving a Matukuma-type source S_(k)(D^(2)(-φ))=|x|^(λ-2)/(1+|x|^(2))^(λ2)φ^(q),x∈Ω,where S_(k)(D^(2)(-φ))is the k-Hessian operator,q>k>1,λ>0,n>2k,k∈N,andΩis a suitable bounded do-main in R~n.It turns out that there are two different types of radially positive solutions for k>1,i.e.,M-solution(singular at r=0)and Esolution(regular at r=0),which is distinct from the case when k=1.For 1<q<[(n-2+λ)k](n-2k),we apply an iterative approach to improve accuracy of asymptotic expansions of M-solution step by step to the desired extend.In contrast to the case k=1,we require a more precise range of parameters due to repeated application of Taylor expansions,which also makes asymptotic expansions need more delicate investigation.
基金TheKeyProjectofChineseMinistryofEducation (No .10 40 90 ) .
文摘This paper investigates the self-similar singular solution of the p-Laplacian evolution equation with the nonlinear gradient absorption terms u t=div(u p-2u)-uq for 1<p<2 and q>in Rn× (0, ∞). It has been proved that when 1<q<p-n/(n+1) there exists a unique self-similar very singular solution.
基金Project supported by the National Natural Science Foundation of China (No. 10102019).
文摘In the present paper, closed form singular solutions for an infinite anisotropic plate with an elliptic hole or crack are derived based on the Stroh-type formalism for the general anisotropic plate. With the solutions, the hoop stresses and hoop moments around the elliptic hole as well as the stress intensity factors at the crack tip under concentrated in-plane stresses and bending moments are obtained. The singular solutions can be used for approximate analysis of an anisotropic plate weakened by a hole or a crack under concentrated forces and moments. They can also be used as fundamental solutions of boundary integral equations in BEM analysis for anisotropic plates with holes or cracks under general force and boundary conditions.
基金The work of the author has been supported by the Deutache Forschungsgemeinschaft(DFG) under Grant Ho 1846/1-1
文摘This paper studies several problems , which are potentially relevant for the construction of adaptive numerical schemes. First, biorthogonal spline wavelets on [0,1] are chosen as a starting point for characterizations of functions in Besom spaces B(?)(0,1) with 0<σ<∞ and (1+σ)-1<γ<∞. Such function spaces are known to be related to nonlinear approximation. Then so called restricted nonlinear approximation procedures with respect to Sobolev space norms are considered. Besides characterization results Jackson type estimates for various tree-type and tresholding algorithms are investigated. Finally known approximation results for geometry induced singularity functions of boundary integeral equations are combined with the characterization results for restricted nonlinear approximation to show Besov space regularity results.
基金Project supported by the National Natural Science Foundation of China (No. 10471022)the Science and Technology Foundation of Ministry of Education of China (Major Projects) (No.104090)
文摘The self-similar singular solution of the fast diffusion equation with nonlinear gradient absorption terms are studied. By a self-similar transformation, the self-similar solutions satisfy a boundary value problem of nonlinear ordinary differential equation （ODE）. Using the shooting arguments, the existence and uniqueness of the solution to the initial data problem of the nonlinear ODE are investigated, and the solutions are classified by the region of the initial data. The necessary and sufficient condition for the existence and uniqueness of self-similar very singular solutions is obtained by investigation of the classification of the solutions. In case of existence, the self-similar singular solution is very singular solution.
基金Supported by the National Natural Science Foundation of China under Grant Nos 11571366 and 11501570the Open Foundation of State Key Laboratory of High Performance Computing of China+1 种基金the Research Fund of the National University of Defense Technology under Grant No JC15-02-02the Fund from HPCL
文摘The two-component Camassa–Holm equation includes many intriguing phenomena. We propose a multi-symplectic compact method to solve the two-component Camassa–Holm equation. Based on its multi-symplectic formulation, the proposed method is derived by the sixth-order compact finite difference method in spatial discretization and the symplectic implicit midpoint scheme in temporal discretization. Numerical experiments finely describe the velocity and density variables in the two-component integrable system and distinctly display the evolvement of the singular solutions. Moreover, the proposed method shows good conservative properties during long-time numerical simulation.
基金B. Yuan was partially supported by the China postdoctoral Science Foundation （No. 20060390530）, Natural Science Foundation of Henan Province （No. 0611055500）, Science Foundation of the Education Department of Henan Province （200510460008） and Doctor Foundation of Henan Polytechnic University.
文摘This paper studies the Cauchy problem of the dissipative quasi-geostrophic equation in pseudomeasure space PM^n＋1-2α（R^n） or Lorentz space L n/2α-1,∞（R^n）, which admit the singular solutions. The global well-posedness is established provided initial data θ0（x） are small enough in these spaces. Moreover, the asymptotic stability of solutions in pseudomeasure space is proved. In particular, if the initial data are homo-geneous functions of degree 1 - 2α, the self-similar solutions are also obtained.
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=1ci｜x｜liupi=0,x∈R^N are obtained, wheren≥3, ci 〉0, li〉-2, i=1,2,..,k, pi〉 1, i=l,2,-..,kandthe 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））.
文摘We study overdetermined systems of first order partial differential equations with singular solutions.The main result gives a characterization of such systems and asserts that the singular solution is equal to the contact singular set.
文摘The case of a radial initial state for a family of hyperbolic systems of con- servation laws with several spatial dimensions is considered. It will be shown that the singularity at the origin introduces multiple solutions outside of the traditional admissible classes.
基金The authors would thank Professor Li Yi very much for helpful discussion. Zhang is supported by the National Natural Science Foundation of China （No. 11371286, 11401458） and by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.
文摘We revise one case of the M-solutions of B. Wang, et al., J. Diff. Eqs. 253 （2012）, pp. 3232-3265 and obtain more precise form of the singular term and the regular term.
文摘In this paper, we investigate the existence of positive solutions for the singular fourth-order differential system <em>u</em><sup>(4)</sup> = <em><span style="white-space:nowrap;">φ</span>u</em> + <em>f </em>(<em>t</em>, <em>u</em>, <em>u</em>”, <em><span style="white-space:nowrap;">φ</span></em>), 0 < <em>t</em> < 1, -<em><span style="white-space:nowrap;">φ</span></em>” = <em>μg</em> (<em>t</em>, <em>u</em>, <em>u</em>”), 0 < <em>t</em> < 1, <em>u</em> (0) = <em>u</em> (1) = <em>u</em>”(0) = <em>u</em>”(1) = 0, <em><span style="white-space:nowrap;">φ</span> </em>(0) = <em><span style="white-space:nowrap;">φ</span> </em>(1) = 0;where <em>μ</em> > 0 is a constant, and the nonlinear terms<em> f</em>, <em>g</em> may be singular with respect to both the time and space variables. The results obtained herein generalize and improve some known results including singular and non-singular cases.
基金Foundation item is supported by the NNSF of China(19971064)
文摘In this paper, the difficulties on calculation in solving singular integral equations are overcome when the restriction of curve of integration to be a closed contour is cancelled. When the curve is an open arc and the solutions for singular integral equations possess singularities of higher order, the solution and the solvable condition for characteristic equations as well as the generalized Noether theorem for complete equations are given.
基金Supported by the National Natural Science Foundation of China(19971064)Ziqiang Invention Foundation of Wuhan University(201990336)
文摘The basic sets of solutions in classH(orH*)for the characteristic equation and its adjoint equation with Hilbert kernel are given respectively.Thus the expressions of solutions and its solvable conditions are simplified.On this basis the solutions and the solvable conditions in classH_(1)as well as the generalized Noether theorem for the complete equation are obtained.
基金Supported by the NNSF of China (10471107)RFDP of Higher Education of China (20060486001)
文摘We transform the singular integral equations with solutions simultaneously having singularities of higher order at infinite point and at several finite points on the real axis into ones along a closed contour with solutions having singularities of higher order, and for the former obtain the extended Neother theorem of complete equation as well as the solutions and the solvable conditions of characteristic equation from the latter. The conclusions drawn by this article contain special cases discussed before.
基金Supported by the National Natural Science Foundation of China (10471107)the Specialized Research Fund for the Doctoral Program of Higher Education of China (20060486001)
文摘Under the appropriate hypotheses subject to the unknown function and the free term, by means of our Lemma, we prove the rationality of order at x = ∞ on two sides for the characteristic singular integral equations with solutions having singularities of higher order on the real axis X. We transform the equations into solving equivalent Riemann boundary value problems with solutions having singularities of higher order and with additional conditions on X. The solutions and the solvable conditions for the former are obtained from the latter.
文摘In this paper we consider the singularly perturbed boundary value problem for the fourth-order elliptic differential equation, establish the energy estimates of the solutionand its derivatives and construct the formal asymptotic solution by Lyuternik- Vishik 's method. Finally, by means of the energy estimates we obtain the bound of the remainder of the asymptotic solution.
基金Supported by the Scientific Research Funds for the Ningxia Universities(Grant No.NGY2015141)
文摘We establish the existence of positive periodic solutions of the second-order singular coupled systems{x′′＋ p_1（t）x′＋ q_1（t）x = f_1（t, y（t）） ＋ c_1（t）,y′′＋ p_2（t）y′＋ q_2（t）y = f_2（t, x（t）） ＋ c_2（t）,where pi, qi, ci ∈ C（R/T Z; R）, i = 1, 2; f_1, f_2 ∈ C（R/T Z ×（0, ∞）, R） and may be singular near the zero. The proof relies on Schauder＇s fixed point theorem and anti-maximum principle.Our main results generalize and improve those available in the literature.
