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In this paper, we consider the local existence of solutions to the Cauchy problems for the following nonlinear evolution equations with mixed types t D 1 / x C xx; t D 1 / C x C 2 x C xx; with initial data . ; /.x; 0/ D . 0.x/; 0....
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In this paper, we consider the local existence of solutions to the Cauchy problems for the following nonlinear evolution equations with mixed types t D 1 / x C xx; t D 1 / C x C 2 x C xx; with initial data . ; /.x; 0/ D . 0.x/; 0.x// ! . ; /; as x ! 1; where and are positive constants satisfying < 1, < .1/. Through constructing an approximation solution sequence, we obtain the local existence by using the contraction mapping principle.
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Of concern is the nonlinear evolution equation in a real Banach space X, where the nonlinear, time-dependent, multi-valued operator A(t): D(A(t)) X → X has a time-dependent domain D(A(t)). It will be shown that, under some non-dis...
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Of concern is the nonlinear evolution equation in a real Banach space X, where the nonlinear, time-dependent, multi-valued operator A(t): D(A(t)) X → X has a time-dependent domain D(A(t)). It will be shown that, under some non-dissipativity condition, the equation has a strong solution. Illustrations are given of solving quasi-linear partial differential equations of parabolic type.
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In this paper, we study the existence and uniqueness of solutions for several classes of stochastic evolution equations with non-Lipschitz coefficients, that contains backward stochastic evolution equations, stochastic Volterra ty...
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In this paper, we study the existence and uniqueness of solutions for several classes of stochastic evolution equations with non-Lipschitz coefficients, that contains backward stochastic evolution equations, stochastic Volterra type evolution equations and stochastic functional evolution equations. In particular, the results can be used to treat a large class of quasi-linear stochastic equations, which includes the reaction diffusion and porous medium equations.
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In this work a class of self-adjoint quasilinear third-order evolution equations is determined. Some conservation laws of them are established and a generalization on a self-adjoint class of fourth-order evolution equations is presented.
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This paper gives two new families of nonlinear partial differential equations (PDEs). One has cusp soliton solution while the other possesses the cusp-like singular traveling wave solution. A typical integrable system: Harry-Dym (...
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This paper gives two new families of nonlinear partial differential equations (PDEs). One has cusp soliton solution while the other possesses the cusp-like singular traveling wave solution. A typical integrable system: Harry-Dym (HD) equation is able to be contained in both families and has cusp soliton solution as well as cusp-like singular traveling wave solution. We prove that the cusp solution of the HD equation is not stable and the cusp-like solution is not included in the parametric solutions of the HD equation. (c) 2004 Elsevier Ltd. All rights reserved.
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Consider the following infinite dimensional stochastic evolution equation over some Hilbert space H with norm /./: X-t = x(0) + integral(o)(t) f(X-s, s) ds + integral(0)(t) g(X-s, s) dW(s), t greater than or equal to 0, P almost s...
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Consider the following infinite dimensional stochastic evolution equation over some Hilbert space H with norm /./: X-t = x(0) + integral(o)(t) f(X-s, s) ds + integral(0)(t) g(X-s, s) dW(s), t greater than or equal to 0, P almost surely. It is proved that under certain mild assumptions, the strong solution X-t(x(0)) epsilon V --> H --> V*, t greater than or equal to 0, is mean square exponentially stable if and only if there exists a Lyapunov functional Lambda(. , .): H x R* --> R-1 which satisfies the following conditions: (i) c(1)/x/(2) - k(1) e-(mu 1t) less than or equal to Lambda(x, t) less than or equal to c(2)/x/(2) + k(2)e(-mu 2t); (ii) L Lambda(x, t) less than or equal to c(3) Lambda(x, t) + k(3)e(-mu 3t), For All x epsilon V, t greater than or equal to 0; where L is the infinitesimal generator of the Markov process X-t and c(i),k(i),mu(i), i = 1,2,3, are positive constants. As a by-product, the characterization of exponential ultimate boundedness of the strong solution is established as the null decay rates (that is, mu(i) = 0) are considered. [References: 14]
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In a finite-dimensional Hilbert space, we consider the Cauchy problem for a second-order integro-differential evolution equation with memory where the integrand is the product of a difference kernel by a linear operator of the tim...
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In a finite-dimensional Hilbert space, we consider the Cauchy problem for a second-order integro-differential evolution equation with memory where the integrand is the product of a difference kernel by a linear operator of the time derivative of the solution. The main difficulties in finding the approximate value of the solution of such nonlocal problems at a given point in time are due to the need to work with approximate values of the solution for all previous points in time. A transformation of the integro-differential equation in question to a system of weakly coupled local evolution equations is proposed. It is based on the approximation of the difference kernel by a sum of exponentials. We state a local problem for a weakly coupled system of equations with additional ordinary differential equations. To solve the corresponding Cauchy problem, stability estimates of the solution with respect to the initial data and the right-hand side are given. The main attention is paid to the construction and stability analysis of three-level difference schemes and their computational implementation.
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In this paper we investigate local and global existence as well as asymptotic behavior of the solution for a class of abstract; (hyperbolic) quasilinear equations perturbed by bounded delay operators. We assume that the leading op...
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In this paper we investigate local and global existence as well as asymptotic behavior of the solution for a class of abstract; (hyperbolic) quasilinear equations perturbed by bounded delay operators. We assume that the leading operator is of bounded variation in time. In the last section, the abstract results are applied on a heat conduction model.
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We derive a thermo-hydrodynamic theory for particles and energy flow, based on a nonequilibrium grand-canonical ensemble formalism. The time-dependent kinetic coefficients are explicitly given in terms of microscopic mechanical qu...
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We derive a thermo-hydrodynamic theory for particles and energy flow, based on a nonequilibrium grand-canonical ensemble formalism. The time-dependent kinetic coefficients are explicitly given in terms of microscopic mechanical quantities. The time evolution equations describing the coupled flow of energy and particles are derived. The second-rank tensorial fluxes of current of energy and particles present in the nonequilibrium ensemble are nondiagonal. We obtain a generalized Fick's Law, which presents the effect of the energy flow on the particle diffusion equation. (C) 2004 American Institute of Physics. [References: 15]
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A new method to solve the nonlinear evolution equations is presented, which combines the two kind methods C the tanh function method and symmetry group method. To demonstrate the method, we consider the (2 + 1)-dimensional cubic n...
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A new method to solve the nonlinear evolution equations is presented, which combines the two kind methods C the tanh function method and symmetry group method. To demonstrate the method, we consider the (2 + 1)-dimensional cubic nonlinear Schrodinger (NLS) equation. As a result, some novel solitary solutions of the Schrodinger equation are obtained. And graphs of some solutions are displayed.
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