Blow-up and global solubility in the classical~sense of~the~Cauchy~problem~for~a~formally~hyperbolic~equation with a~non-coercive source

M. O. Korpusov

Izvestiya. Mathematics , Tome 84 (2020) no. 5, pp. 930-959.

Résumé

We consider an abstract Cauchy problem with non-linear operator coefficients and prove the existence of a unique non-extendable classical solution. Under certain sufficient close-to-necessary conditions, we obtain finite-time blow-up conditions and upper and lower bounds for the blow-up time. Moreover, under certain sufficient close-to-necessary conditions, we obtain a result on the existence of a global-in-time solution independently of the size of the initial functions.

Détail

Keywords: non-linear Sobolev-type equations, blow-up, local solubility, non-linear capacity, bounds for the blow-up time.

Bibliographie

[1] H. A. Levine, “Some nonexistence and instability theorems for solutions of formally parabolic equations of the form $Pu_t=-Au+\mathscr F(u)$”, Arch. Rational. Mech. Anal., 51 (1973), 371–386 | DOI | MR | Zbl

[2] H. A. Levine, “Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_{tt}=-Au+\mathscr F(u)$”, Trans. Amer. Math. Soc., 192 (1974), 1–21 | DOI | MR | Zbl

[3] B. Straughan, “Further global nonexistence theorems for abstract nonlinear wave equations”, Proc. Amer. Math. Soc., 48:2 (1975), 381–390 | DOI | MR | Zbl

[4] V. K. Kalantarov, O. A. Ladyzhenskaya, “The occurrence of collapse for quasilinear equations of parabolic and hyperbolic types”, J. Soviet Math., 10:1 (1978), 53–70 | DOI | MR | Zbl

[5] H. A. Levine, J. Serrin, “Global nonexistence theorems for quasilinear evolution equations with dissipation”, Arch. Rational Mech. Anal., 137:4 (1997), 341–361 | DOI | MR | Zbl

[6] H. A. Levine, Sang Ro Park, J. Serrin, “Global existence and nonexistence theorems for quasilinear evolution equations of formally parabolic type”, J. Differential Equations, 142:1 (1998), 212–229 | DOI | MR | Zbl

[7] P. Pucci, J. Serrin, “Some new results on global nonexistence for abstract evolution with positive initial energy”, Topol. Methods Nonlinear Anal., 10:2 (1997), 241–247 | DOI | MR | Zbl

[8] P. Pucci, J. Serrin, “Global nonexistence for abstract evolution equations with positive initial energy”, J. Differential Equations, 150:1 (1998), 203–214 | DOI | MR | Zbl

[9] H. A. Levine, G. Todorova, “Blow up of solutions of the Cauchy problem for a wave equation with nonlinear damping and source terms and positive initial energy”, Proc. Amer. Math. Soc., 129:3 (2001), 793–805 | DOI | MR | Zbl

[10] M. O. Korpusov, “Blow-up of solutions of strongly dissipative generalized Klein–Gordon equations”, Izv. Math., 77:2 (2013), 325–353 | DOI | DOI | MR | Zbl

[11] M. O. Korpusov, “Solution blow-up in a nonlinear system of equations with positive energy in field theory”, Comput. Math. Math. Phys., 58:3 (2018), 425–436 | DOI | DOI | MR | Zbl

[12] B. A. Bilgin, V. K. Kalantarov, “Non-existence of global solutions to nonlinear wave equations with positive initial energy”, Commun. Pure Appl. Anal., 17:3 (2018), 987–999 | DOI | MR | Zbl

[13] V. Georgiev, G. Todorova, “Existence of a solution of the wave equation with nonlinear damping and source terms”, J. Differential Equations, 109:2 (1994), 295–308 | DOI | MR | Zbl

[14] S. A. Messaoudi, “Blow up and global existence in a nonlinear viscoelastic wave equation”, Math. Nachr., 260 (2003), 58–66 | DOI | MR | Zbl

[15] S. A. Messaoudi, “Blow-up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic equation”, J. Math. Anal. Appl., 320:2 (2006), 902–915 | DOI | MR | Zbl

[16] M. O. Korpusov, “Blow-up of solutions of a class of strongly non-linear equations of Sobolev type”, Izv. Math., 68:4 (2004), 783–832 | DOI | DOI | MR | Zbl

[17] M. O. Korpusov, A. G. Sveshnikov, “Blow-up of solutions of a class of strongly non-linear dissipative wave equations of Sobolev type with sources”, Izv. Math., 69:4 (2005), 733–770 | DOI | DOI | MR | Zbl

[18] M. O. Korpusov, A. A. Panin, “Blow-up of solutions of an abstract Cauchy problem for a formally hyperbolic equation with double non-linearity”, Izv. Math., 78:5 (2014), 937–985 | DOI | DOI | MR | Zbl

[19] A. B. Al'shin, M. O. Korpusov, A. G. Sveshnikov, Blow-up in nonlinear Sobolev type equations, De Gruyter Ser. Nonlinear Anal. Appl., 15, Walter de Gruyter Co., Berlin, 2011, xii+648 pp. | DOI | MR | Zbl

[20] M. I. Rabinovich, “Avtokolebaniya raspredelennykh sistem”, Izv. vuzov. Radiofizika, 17:4 (1974), 477–510

[21] D. C. Kim, “Video solitons in a dispersive transmission line with the nonlinear capacitance of a $p-n$-junction”, Tech. Phys., 58:3 (2013), 340–350 | DOI

[22] V. M. Zhuravlev, “Autowaves in double-wire lines with the exponential-type nonlinear active element”, JETP Letters, 75:1 (2002), 9–14 | DOI

[23] M. O. Korpusov, “Destruction of solutions of wave equations in systems with distributed parameters”, Theoret. and Math. Phys., 167:2 (2011), 577–583 | DOI | DOI | MR | Zbl

[24] M. O. Korpusov, “Solution blowup for nonlinear equations of the Khokhlov–Zabolotskaya type”, Theoret. and Math. Phys., 194:3 (2018), 347–359 | DOI | DOI | MR | Zbl

[25] M. O. Korpusov, D. V. Lukyanenko, A. A. Panin, G. I. Shlyapugin, “On the blow-up phenomena for a 1-dimensional equation of ion-sound waves in a plasma: analytical and numerical investigation”, Math. Methods Appl. Sci., 41:8 (2018), 2906–2929 | DOI | MR | Zbl

[26] M. O. Korpusov, D. V. Lukyanenko, “Instantaneous blow-up versus local solvability for one problem of propagation of nonlinear waves in semiconductors”, J. Math. Anal. Appl., 459:1 (2018), 159–181 | DOI | MR | Zbl

[27] M. O. Korpusov, D. V. Lukyanenko, A. A. Panin, E. V. Yushkov, “Blow-up of solutions of a full non-linear equation of ion-sound waves in a plasma with non-coercive non-linearities”, Izv. Math., 82:2 (2018), 283–317 | DOI | DOI | MR | Zbl

[28] A. A. Panin, G. I. Shlyapugin, “Local solvability and solution blow-up of one-dimensional equations of the Yajima–Oikawa–Satsuma type”, Theoret. and Math. Phys., 193:2 (2017), 1561–1573 | DOI | DOI | MR | Zbl

[29] M. O. Korpusov, A. A. Panin, “On the nonextendable solution and blow-up of the solution of the one-dimensional equation of ion-sound waves in a plasma”, Math. Notes, 102:3 (2017), 350–360 | DOI | DOI | MR | Zbl

[30] M. O. Korpusov, D. V. Lukyanenko, E. A. Ovsyannikov, A. A. Panin, “Lokalnaya razreshimost i razrushenie resheniya odnogo uravneniya s kvadratichnoi nekoertsitivnoi nelineinostyu”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 10:2 (2017), 107–123 | Zbl

[31] M. O. Korpusov, D. V. Lukyanenko, A. A. Panin, E. V. Yushkov, “Blow-up phenomena in the model of a space charge stratification in semiconductors: analytical and numerical analysis”, Math. Methods Appl. Sci., 40:7 (2017), 2336–2346 | DOI | MR | Zbl

[32] D. V. Lukyanenko, A. A. Panin, “Razrushenie resheniya uravneniya stratifikatsii ob'emnogo zaryada v poluprovodnikakh: chislennyi analiz pri svedenii iskhodnogo uravneniya k differentsialno-algebraicheskoi sisteme”, Vych. met. programmirovanie, 17:4 (2016), 437–446

[33] A. A. Panin, “Local solvability and blowup of the solution of the Rosenau–Bürgers equation with different boundary conditions”, Theoret. and Math. Phys., 177:1 (2013), 1361–1376 | DOI | DOI | MR | Zbl

[34] M. O. Korpusov, A. A. Panin, “Local solvability and solution blowup for the Benjamin–Bona–Mahony–Burgers equation with a nonlocal boundary condition”, Theoret. and Math. Phys., 175:2 (2013), 580–591 | DOI | DOI | MR | Zbl

[35] M. O. Korpusov, D. V. Lukyanenko, A. A. Panin, E. V. Yushkov, “Blow-up for one Sobolev problem: theoretical approach and numerical analysis”, J. Math. Anal. Appl., 442:2 (2016), 451–468 | DOI | MR | Zbl

[36] È. Mitidieri, S. I. Pokhozhaev, “A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities”, Proc. Steklov Inst. Math., 234 (2001), 1–362 | MR | Zbl

[37] H. Gajewski, K. Gröger, K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Math. Lehrbucher und Monogr., 38, Akademie-Verlag, Berlin, 1974, ix+281 pp. | MR | MR | Zbl

[38] A. A. Panin, “On local solvability and blow-up of solutions of an abstract nonlinear Volterra integral equation”, Math. Notes, 97:6 (2015), 892–908 | DOI | DOI | MR | Zbl

[39] V. I. Bogachev, O. G. Smolyanov, Deistvitelnyi i funktsionalnyi analiz, NITs “Regulyarnaya i khaoticheskaya dinamika”, M.–Izhevsk, 2009, 724 pp.

[40] H. Cartan, Calcul différentiel, Hermann, Paris, 1967, 178 pp. ; Formes différentielles. Applications élémentaires au calcul des variations et à la théorie des courbes et des surfaces, Hermann, Paris, 1967, 186 pp. | MR | MR | MR | Zbl | Zbl

[41] B. P. Demidovich, Lektsii po matematicheskoi teorii ustoichivosti, Nauka, M., 1967, 472 pp. | MR | Zbl