高维非线性热弹方程解的爆破
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摘要
本文研究下面给出的一类有外力作用和热存储的高维非线性热弹方程的初边值问题弱解的爆破现象：这里u=u(x，t)表示位移；θ=θ(x，t)表示温度差；Ω是R~n(n≥1)上的有光滑边界(?)Ω的有界区域；函数f=f(t，u)是一个非自治外力；函数g(t)是松弛核函数且*表示卷积，即g*y(·，t)=integral from n=0 to t g(t-Υ)y(·，Υ)dΥ；u_0(x)，u_1(x)，θ_0(x)表示给定的初值，并且满足相容性条件。与一般的热弹方程组相比，我们所考虑的方程组含有核项g*[divK_2▽θ]，我们主要研究了当g(t)为正定核，或者由g(t)构造的一类函数e~(αt)g(t)和e~(-αt)g(t)(α＞0)是正定核时，上述问题的解的爆破情况。
In this paper we study the blow-up phenomena of weak solutions in a finite time to the following initial boundary value problem with a forcing term and a thermal memory for multi—dimensional thermoelastic model :
Here by u = u(x, t) and θ = θ(x, t) we denote the displacement and the temperature difference, resperctively; Ω is a bounded domain in R~n(n ≥ 1)with a smooth boundary (?)Ω ; the function f = f(t,u) is a nonautonomous forcing term ; the function g — g(t) is the relaxation kernel and the sign * denotes the convolution product , i.e. , g* y(.,t) = ∫_0~t g(t ~Τ)y(.,T)dT ; u_0(x),u_1(x),θ_0(x) denotes the initial date . we consider the problem with the kernel term g*[divK_2▽θ] ,and prove the blow—up phenomena of weak solutions in a finite time with g(t) is a positive definite kernel or with the positive kernels of the forms e~(αt)g(t) and e~~(αt)g(t)(α> 0) constructed from g(t) .
In this paper we study the blow-up phenomena of weak solutions in a finite time to the following initial boundary value problem with a forcing term and a thermal memory for multi—dimensional thermoelastic model :
Here by u = u(x, t) and θ = θ(x, t) we denote the displacement and the temperature difference, resperctively; Ω is a bounded domain in R~n(n ≥ 1)with a smooth boundary (?)Ω ; the function f = f(t,u) is a nonautonomous forcing term ; the function g — g(t) is the relaxation kernel and the sign * denotes the convolution product , i.e. , g* y(.,t) = ∫_0~t g(t ~Τ)y(.,T)dT ; u_0(x),u_1(x),θ_0(x) denotes the initial date . we consider the problem with the kernel term g*[divK_2▽θ] ,and prove the blow—up phenomena of weak solutions in a finite time with g(t) is a positive definite kernel or with the positive kernels of the forms e~(αt)g(t) and e~~(αt)g(t)(α> 0) constructed from g(t) .
引文
[1] C.M.Dafermos, On the existence and the asymptotic stability of solutions to the equtions of linear thermoelasticity, Arch.Rational Mech.Anal.6(1979)241-271.
[2] J.E.M.Rivera. Energy decay rates in linear thermoelasticity. Funkcial Ekvac.35(1992)19-35.
[3] J.E.M.Rivera. Y.Qin, Global existence and exponential stability in one—dimensional nonlinear thermoelasticity with thermal memory, Nonlinear Anal.51(2002) 11—32.
[4] M.Kirane, N.Tatar, A nonexistence result to a cauchy problem in nonlinear one-dimensional thermoelasticity, J.Math.Anal.Appl.254(2001)71-86.
[5] M.Slemrod, Global existence, uniqueness, and asymptotic stability of classical smooth solutions in one-dimensional nonlinear thermoelasticity, Arch.Rational Mech.Anal.76(1981)97-133.
[6] M.Kirane, S.Kouachi, N.Tatar. Nonlinear of global solusions of some quasilinear hyperbolic equations with dynamic boundary conditions, Math.Nachr. 176(1995)139-147.
[7] R.Racke, Y.Shibata, S.Zheng,Global solvability and exponential stability in one—dimensional nonlinear thermoelasticity, Quart.Appl.Math.51(1993)751-763.
[8] R.Racke, Y.Shibata, Global smooth solutions and asymptotic stability in one—dimensional nonlinear thermoelasticity, Arch.Rational Mech.Anal. 116(1991) 1-34.
[9] R.Racke, On the Cauchy problem in 3-d thermoelasticity, Math.Z. 203(1990)649-682.
[10] R.Racke, Lectures on nonlinear evolution equtions, initial value problems, Vieweg Sohn, braunschweig/Wiesbaden 1992.
[11] S.Zheng, Global solution and application to a class of quasilinear hyperbolic-parabolic coupled system, Sci.Sinica Ser.A27(1984)1274-1286.
[12] S.Zheng, W.Shen, Global solutions to the Cauchy problem of a class of hyperbolic—parabolic coupled system, Sci.Sinica Ser.A30(1987) 1133-1149.
[13] S.Jiang, On global smooth solutions to the one—dimensional equations of nonlinear inhomo- geneous thermoelasticity. Nonlinear Anal. 20(1993)1245-1256.
[14] S.M.Zheng, W.X.Shen, Global solusions to the Cauchy problem of equations of one—dimensional thermoelasticity, J Partial Diff.Eqs.2(1989)26-38.
[15] S.A.Messaoudi, A blowup result in a multidimensional semilinear thermoelastic system, Electronic J, Differential Equtions 30(2001)1-9.
[16] S.Jiang, R.Racke, Evolution equations in thermoelasticity, Chapman Hall/CRC Monographs and Survers in Pure and Appl. Math. Vol.112, Chapman Hall/CRC 2000.
[17] T.Mukoyama, On the Cauchy problem for quasilinear hyperbolic parabolic coupled systems in higher dimensional spaces, Tsukuba J. Math. 13(1989)363-386.
[18] V.K.Kalantanrov, O.A.Ladyzhenskaya, The occurrence of collapse for quasilinear equtions of parabolic and hyperbolic types, J.Soviet Math.10(1978)53-70.
[19] Y.Qin, J.M.Rivera, Blow-up of solutions to the Cauchy problem in nonlinear one-dimensional thermoelasticity, J.Math.Anal.Appl.292(2004)160-193.
[20] Y.Qin, Global existence and asymptotic behaviour of solutions to a system of equations for a nonlinear one-dimensional viscous heat-conducting real gas, Chinese Ann. Math. 20A 3(1999)343-354.
[21] Y.Qin, Global existence and asymptotic behaviour of solutions to nonlinear hyperbolic-parabolic coupled systems with arbitrary initial data . Ph.D. Thesis, Fudan University, 1998.
[22] Y.Qin, Global existence and asymptotic behaviour for the solutions to nonlinear viscous, heat-conductive,one-dimension real gas, Adv.Math.Sci.Appl.10(2000)119-148.
[23] Y.Qin, Global existence and asymptotic behaviour for a viscous,heat-conductive, one-dimensional real gas with fixed and thermally insulated endpoints, Nonlinear Anal. Theoty Methods Appl. 44(2001)413-441.
[24] Y.Qin. Global existence and asymptotic behaviour of solition to the system in one-dimensional nonlinear thermoviscoelasticity, Quart . Appl . Math . 59(2001)113-142.
[25] Y.Qin, Global existence and asymptotic behaviour for a viscous heat-conductive, one-dimensional real gas with fixed and constant temperature boundary conditions, Adv. Differential Equatons 7(2002)129-154.
[2] J.E.M.Rivera. Energy decay rates in linear thermoelasticity. Funkcial Ekvac.35(1992)19-35.
[3] J.E.M.Rivera. Y.Qin, Global existence and exponential stability in one—dimensional nonlinear thermoelasticity with thermal memory, Nonlinear Anal.51(2002) 11—32.
[4] M.Kirane, N.Tatar, A nonexistence result to a cauchy problem in nonlinear one-dimensional thermoelasticity, J.Math.Anal.Appl.254(2001)71-86.
[5] M.Slemrod, Global existence, uniqueness, and asymptotic stability of classical smooth solutions in one-dimensional nonlinear thermoelasticity, Arch.Rational Mech.Anal.76(1981)97-133.
[6] M.Kirane, S.Kouachi, N.Tatar. Nonlinear of global solusions of some quasilinear hyperbolic equations with dynamic boundary conditions, Math.Nachr. 176(1995)139-147.
[7] R.Racke, Y.Shibata, S.Zheng,Global solvability and exponential stability in one—dimensional nonlinear thermoelasticity, Quart.Appl.Math.51(1993)751-763.
[8] R.Racke, Y.Shibata, Global smooth solutions and asymptotic stability in one—dimensional nonlinear thermoelasticity, Arch.Rational Mech.Anal. 116(1991) 1-34.
[9] R.Racke, On the Cauchy problem in 3-d thermoelasticity, Math.Z. 203(1990)649-682.
[10] R.Racke, Lectures on nonlinear evolution equtions, initial value problems, Vieweg Sohn, braunschweig/Wiesbaden 1992.
[11] S.Zheng, Global solution and application to a class of quasilinear hyperbolic-parabolic coupled system, Sci.Sinica Ser.A27(1984)1274-1286.
[12] S.Zheng, W.Shen, Global solutions to the Cauchy problem of a class of hyperbolic—parabolic coupled system, Sci.Sinica Ser.A30(1987) 1133-1149.
[13] S.Jiang, On global smooth solutions to the one—dimensional equations of nonlinear inhomo- geneous thermoelasticity. Nonlinear Anal. 20(1993)1245-1256.
[14] S.M.Zheng, W.X.Shen, Global solusions to the Cauchy problem of equations of one—dimensional thermoelasticity, J Partial Diff.Eqs.2(1989)26-38.
[15] S.A.Messaoudi, A blowup result in a multidimensional semilinear thermoelastic system, Electronic J, Differential Equtions 30(2001)1-9.
[16] S.Jiang, R.Racke, Evolution equations in thermoelasticity, Chapman Hall/CRC Monographs and Survers in Pure and Appl. Math. Vol.112, Chapman Hall/CRC 2000.
[17] T.Mukoyama, On the Cauchy problem for quasilinear hyperbolic parabolic coupled systems in higher dimensional spaces, Tsukuba J. Math. 13(1989)363-386.
[18] V.K.Kalantanrov, O.A.Ladyzhenskaya, The occurrence of collapse for quasilinear equtions of parabolic and hyperbolic types, J.Soviet Math.10(1978)53-70.
[19] Y.Qin, J.M.Rivera, Blow-up of solutions to the Cauchy problem in nonlinear one-dimensional thermoelasticity, J.Math.Anal.Appl.292(2004)160-193.
[20] Y.Qin, Global existence and asymptotic behaviour of solutions to a system of equations for a nonlinear one-dimensional viscous heat-conducting real gas, Chinese Ann. Math. 20A 3(1999)343-354.
[21] Y.Qin, Global existence and asymptotic behaviour of solutions to nonlinear hyperbolic-parabolic coupled systems with arbitrary initial data . Ph.D. Thesis, Fudan University, 1998.
[22] Y.Qin, Global existence and asymptotic behaviour for the solutions to nonlinear viscous, heat-conductive,one-dimension real gas, Adv.Math.Sci.Appl.10(2000)119-148.
[23] Y.Qin, Global existence and asymptotic behaviour for a viscous,heat-conductive, one-dimensional real gas with fixed and thermally insulated endpoints, Nonlinear Anal. Theoty Methods Appl. 44(2001)413-441.
[24] Y.Qin. Global existence and asymptotic behaviour of solition to the system in one-dimensional nonlinear thermoviscoelasticity, Quart . Appl . Math . 59(2001)113-142.
[25] Y.Qin, Global existence and asymptotic behaviour for a viscous heat-conductive, one-dimensional real gas with fixed and constant temperature boundary conditions, Adv. Differential Equatons 7(2002)129-154.