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        具有非線性項的弱耦合半線性Moore-Gibson-Thompson系統(tǒng)解的全局非存在性

        2022-05-10 10:26歐陽柏平肖勝中
        關(guān)鍵詞:爆破

        歐陽柏平 肖勝中

        摘要:考慮了一類非線性項的弱耦合半線性Moore-Gibson-Thompson(MGT)系統(tǒng)柯西問題解的爆破現(xiàn)象。在次臨界情況下,運(yùn)用泛函分析和迭代方法推出了其解的全局非存在性。另外,證明了其解的生命跨度的上界估計。

        關(guān)鍵詞:非線性項;Moore-Gibson-Thompson系統(tǒng);爆破

        中圖分類號:O175.4文獻(xiàn)標(biāo)志碼:A

        Moore-Gibson-Thompson(MGT)方程在實際中有廣泛的應(yīng)用[1-3]。物理上,其可描述波在粘性熱松弛流體中的傳播,數(shù)學(xué)模型為

        τu+u-cΔu-bΔu=0

        式中:u為聲速勢函數(shù),u=u(t,x);c為聲速;b為聲擴(kuò)散率,b=βc;τ為松弛因子,τ∈(0,β]。半群理論指出,當(dāng)τ=β時,不存在半群指數(shù)穩(wěn)定性。

        更多半線性MGT方程解的全局存在和爆破問題等解的性態(tài)的相關(guān)研究,請參考文獻(xiàn)[4-12]。

        文獻(xiàn)[13]考慮了如下非線性項的半線性MGT方程解的爆破問題

        在次臨界和臨界2種情況下,作者主要利用迭代技巧和測試函數(shù)方法證明了其柯西問題解的全局非存在性,進(jìn)一步推出了2種情況下其解的生命跨度上界估計。

        一些文獻(xiàn)[14-19]探討了下面具有非線性項的弱耦合半線性波動系統(tǒng)解的爆破問題

        本文目標(biāo)主要是分析弱耦合半線性MGT系統(tǒng)中非線性項對解的爆破以及生命跨度的影響。生命跨度(lifespan)指的是保證解存在的時間區(qū)間最大長度[20]。與近期的工作[13]相比,本文考慮的是非線性弱耦合系統(tǒng)解的爆破。對于式(1)滿足β=β且p=q時,弱耦合問題(1)將一定程度退化為單個的非線性MGT方程;但是當(dāng)p≠q時,本文所研究的模型并不是單個方程的簡單推廣。由于右端弱耦合現(xiàn)象的出現(xiàn),使得臨界曲線的非對稱區(qū)域研究更為復(fù)雜;而對比經(jīng)典的弱耦合波動方程的研究[14-19],由于關(guān)于時間的高階導(dǎo)出現(xiàn),使得無界乘子產(chǎn)生較大作用,同時也使得經(jīng)典的反射法、迭代法均不適用。因此,本文發(fā)展的在弱耦合非線性MGT方程組中研究解的爆破準(zhǔn)則并不是前人工作的簡單推廣。

        另外,由于無界乘子的引入,導(dǎo)致無法應(yīng)用Kato引理研究其解的爆破情況。因此,本文運(yùn)用近年來學(xué)者提出的處理某些高階雙曲方程解的爆破問題的迭代技巧[21-27],輔之以測試函數(shù)和相關(guān)的泛函分析方法進(jìn)行研究。其中,如何選擇適合的測試函數(shù)進(jìn)行迭代是難點(diǎn)。本文通過構(gòu)造恰當(dāng)?shù)哪芰糠汉约袄梦⒎植坏仁郊记傻玫搅似湎陆缧蛄校M(jìn)一步迭代,證明了非臨界情況下具有非線性項的弱耦合半線性MGT系統(tǒng)柯西問題解的全局非存在性,以及解的生命跨度的上界估計。

        1 主要結(jié)果

        首先定義問題(1)的柯西問題能量解:

        參考文獻(xiàn):

        [1]CRIGHTON D G. Model equations of nonlinear acoustics[J]. Annual Review of Fluid Mechanics, 1979, 11(1): 11-33.

        [2] JORDAN P M. Nonlinear acoustic phenomena in viscous thermally relaxing fluids: shock bifurcation and the emergence of diffusive solitons[J]. The Journal of the Acoustical Society of America, 2008, 124(4): 2491.

        [3] KALTENBACHER B, LASIECKA I, MARCHAND R. Well-posedness and exponential decay rates for the Moore-Gibson-Thompson equation arising in high intensity ultrasound[J]. Control and Cybernetics, 2011, 40(4): 971-988.

        [4] ALVES M O, CAIXETA A H, SILVA M A J, et al. Moore-Gibson-Thompson equation with memory in a history framework: a semigroup approach[J]. Zeitschrift fur Angewandte Mathematik und Physik, 2018, 69: 19.

        [5] CAIXETA A H, LASIECKA I, DOMINGOS CAVALCANTI V N. On long time behavior of Moore-Gibson-Thompson equation with molecular relaxation[J]. Evolution Equations and Control Theory, 2016, 5: 661-676.

        [6] DELL′ORO F, LASIECKA I, PATA V. A note on the Moore-Gibson-Thompson equation with memory of type II[J]. Journal of Evolution Equations, 2020, 20: 1251-1268.

        [7] DELL′ORO F, LASIECKA I, PATA V. The Moore-Gibson-Thompson equation with memory in the critical case[J]. Journal of Differential Equations, 2016, 261: 4188-4222.

        [8] DELL′ORO F, PATA V. On the Moore-Gibson-Thompson equation and its relation to linear viscoelasticity[J]. Applied Mathematics and Optimization, 2017, 76: 641-655.

        [9] LASIECKA I. Global solvability of Moore-Gibson-Thompson equation with memory arising in nonlinear acoustics[J]. Journal of Evolution Equations, 2017, 17: 411-441.

        [10]LASIECKA I, WANG X J. Moore-Gibson-Thompson equation with memory, part I: exponential decay of energy[J]. Zeitschrift fur Angewandte Mathematik und Physik, 2016, 67: 23.

        [11]PELLICER M, SAID-HOUARI B. Wellposedness and decay rates for the Cauchy problem of the Moore-Gibson-Thompson equation arising in high intensity ultrasound[J]. Applied Mathematics and Optimization, 2019, 80: 447-478.

        [12]PELLICER M, SOL-MORALES J. Optimal scalar products in the Moore-Gibson-Thompson equation[J]. Evolution Equations and Control Theory, 2019, 8: 203-220.

        [13]CHEN W H, PALMIERI A. Nonexistence of global solutions for the semilinear Moore-Gibson-Thompson equation in the conservative case[J]. Discrete and Continuous Dynamical Systems, 2020, 40: 5513-5540.

        [14]AGEMI R, KUROKAWA Y, H. TAKAMURA H. Critical curve for p-q systems of nonlinear wave equations in three space dimensions[J]. Journal of Differential Equations, 2000, 167(1): 87-133.

        [15]DEL SANTO D, MITIDIERI E. Blow-up of solutions of a hyperbolic system: the critical case[J]. Differential Equations, 1998, 34 (9): 1157-1163.

        [16]GEORGIEV V, TAKAMURA H, ZHOU Y. The lifespan of solutions to nonlinear systems of a high dimensional wave equation[J]. Nonlinear Analysis, 2006, 64(10): 2215-2250.

        [17]KUROKAWA Y. The lifespan of radially symmetric solutions to nonlinear systems of odd dimensional wave equations[J]. Tsukuba Journal of Mathematics, 2005, 60(7): 1239-1275.

        [18]KUROKAWA Y, TAKAMURA H. A weighted pointwise estimate for two dimensional wave equations and its applications to nonlinear systems[J]. Tsukuba Journal of Mathematics, 2003, 27 (2): 417-448.

        [19]KUROKAWA Y, TAKAMURA H, WAKASA K. The blow-up and lifespan of solutions to systems of semilinear wave equation with critical exponents in high dimensions[J]. Differential Integral Equations, 2012, 25(3/4): 363-382.

        [20]李大潛, 周憶. 非線性波動方程[M]. 上海: 上海科學(xué)技術(shù)出版社, 2015.

        [21]CHEN W H, PALMIERI A. A blow-up result for the semilinear Moore-Gibson-Thompson equation with nonlinearity of derivative type in the conservative case[J/OL]. Evolution Equations and Control Theory, (2019-09-20)[2021-06-22].https://arxiv.org/abs/1909.09348.DOI:10.3934/eect.2020085.

        [22]CHEN W H, IKEHATA R. The Cauchy problem for the Moore-Gibson-Thompson equation in the dissipative case[J]. Journal of Differential Equations, 2021, 292: 176-219.

        [23]CHEN W H. Interplay effects on blow-up of weakly coupled systems for semilinear wave equations with general nonlinear memory terms[J]. Nonlinear Analysis, 2021, 202: 112160.1-112160.23

        [24]CHEN W H, REISSIG M. Blow-up of solutions to Nakao′s problem via an iteration argument[J]. Journal of Differential Equations, 2021, 275: 733-756.

        [25]CHEN W H, PALMIERI A. Weakly coupled system of semilinear wave equations with distinct scale-invariant terms in the linear part[J]. Zeitschrift für Angewandte Mathematik und Physik, 2019, 70(2): 67.

        [26]LAI N A, TAKAMURA H, WAKASA K. Blow-up for semilinear wave equations with the scale invariant damping and super-Fujita exponent[J]. Journal of Differential Equations, 2017, 263: 5377-5394.

        [27]PALMIERI A, TAKAMURA A. Blow-up for a weekly coupled system of semilinear damped wave equations in the scattering case with power nonlinearities[J]. Nonlinear Analysis, 2019, 187: 467-492.

        [28]陳國旺. 索伯列夫空間導(dǎo)論[M]. 北京: 科學(xué)出版社, 2013.

        [29]YORDANOV B T, ZHANG Q S. Finite time blow up for critical wave equations in high dimensions[J]. Journal of Functional Analysis, 2006, 231 (2): 361-374.

        [30]LAI N A, TAKAMURA H. Nonexistence of global solutions of nonlinear wave equations with weak time-dependent damping related to Glassey conjecture[J]. Differential Integral Equations, 2019, 32: 37-48.

        (責(zé)任編輯:周曉南)

        Nonexistence of Global Solutions to a Weakly Coupled Semilinear

        Moore-Gibson-Thompson System with Nonlinear Terms

        OUYANG Baiping XIAO Shengzhong

        (1.Guangzhou Huashang College, Guangzhou 511300, China; 2.Guangdong AIB Polytechnic College, Guangzhou 510507, China)Abstract: Blow-up of solutions to the Cauchy problem for a weakly coupled semilinear Moore-Gibson- Thompson(MGT) system with nonlinear terms is considered. Nonexistence of global solutions to the Cauchy problem for the semilinear MGT equation in the subcritical case is derived by applying functional analysis and iteration methods. Additionally, an upper bound estimate of solutions for the lifespan is proved.

        Key words: nonlinear term; Moore-Gibson-Thompson system; blow-up

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