The Absence of Positive Global Periodic Solution of a Second-Order Semi Linear Parabolic Equation With Time-Periodic Coefficients

Year 2024, Volume: 14 Issue: 2, 20 - 35, 31.07.2024

Shirmayil G. Bagyrov

Abstract

Second order semilinear parabolic equation ∂u∂t = div(A(x, t)∇u) + h(x, t, u)
with time-periodic coefficients is considered in domain Ω × (−∞, +∞), where Ω is the
exterior of a compact set in Rnx. Depending on the behavior of the function h(x, t, u)
at infinity, the conditions are found under which the positive periodic solution does not
exist.

Keywords

semilinear parabolic equation, periodic solution, nonexistence of global solutions.

References

• [1] S. Alarcon, J. Melian, A. Quaas, Optimal Liouville theorems for supersolutions of elliptic equations with the Laplacian, Ann Sc Norm Sup Pisa, 16, 2016, 129-158.
• [2] S. Armstrong, B. Sirakov, Sharp Liouville results for fully nonlinear equations with power-growth nonlinearities, Ann Sc Norm Sup Pisa, 10, 2011, 711-728.
• [3] Sh.G. Bagyrov, On the existence of a positive solution of second-order parabolic equations with time-periodic coefficients, Vestnik Moskovskogo Universiteta, Seriya 1, Matematika, Mekhanika, 2, 1996, 86–89.
• [4] Sh.G. Bagyrov, On the existence of a positive solution of a nonlinear secondorder parabolic equation with time-periodic coefficients, Differentsial’nye Uravneniya, 43(4), 2007, 562-565.
• [5] Sh.G. Bagyrov, Absence of positive solutions of a second-order semilinear parabolic equation with time-periodic coefficients, Differential Equations, 50(4), 2014, 1-6.
• [6] Sh.G. Bagyrov, On non-existence of positive periodic solution for second order semilinear parabolic equation, Azerbaijan J. of Math, 8(2), 2018, 163- 180.
• [7] Sh.G. Bagyrov, Nonexistence of global positive solutions of weakly coupled systems of semilinear parabolic equations with time-periodic coefficients, Differential Equations, 56, 2020, 721–733.
• [8] A. Bahri, P.-L. Lions, On the existence of positive solutions of semilinear elliptic equations in unbounded domains, Ann. Inst. H. Poincare Anal. Non Lineaire, 14, 1997, 365-413.
• [9] P. Baras, J.A. Goldstein, The heat equation with a singular potential, Transactions of the American Mathematical Society, 284(1), 1984, 121-139.
• [10] A. Beltramo, P. Hess, On the principal eigenvalue of a periodic-parabolic operator, Communications in Partial Differential Equations, 9(9), 1984, 919- 941.
• [11] M. Bidaut-V’eron, S.Pohozaev, Nonexistence results and estimates for some nonlinear elliptic problems, J D’Analyse Math, 84, 2001, 1-49.
• [12] H. Chen, R. Peng, F. Zhou, Nonexistence of positive super solution to a class of semilinear elliptic equations and systems in an exterior domain, Science China Mathematics, 63(7), 2019, 1-16.
• [13] K. Deng, H.A. Levine, The role of critical exponents in blow-up theorems: the sequel, J. Math. Anal. Appl., 243(1), 2000, 85-126.
• [14] M.J. Esteban, On periodic solutions of superlinear parabolic problems, Transactions of the American Mathematical Society, 293(1), 1986, 171-189
• [15] M.J. Esteban, A remark on the existence of positive periodic solutions of superlinear parabolic problems, Proceedings of the American Mathematical Society, 102(1), 1988, 131-136.
• [16] H. Fujita, On the blowing-up of solutions of the Cauchy problem for ut =∆u + u1+α , J. Fac. Sci. Univ, Tokyo, Sect. I, 13, 1966, 109-124.
• [17] B. Gidas, J. Spruck, Global and local behavior of positive solutions of linear elliptic equations, Comm. Pare. Appl. Math., 34, 1981, 525-598.
• [18] Y. Giga, N. Mizoguchi, On time periodic solutions of the Dirichlet problem for degenerate parabolic equations of nondivergence type, Journal of Mathematical Analysis and Applications, 201(2), 1996, 396-416.
• [19] N. Hirano, N. Mizoguchi, Positive unstable periodic solutions for superlinear parabolic equations, Proceedings of the American Mathematical Society, 123(5), 1995, 1487-1495.
• [20] J. Huska, Periodic solutions in semilinear parabolic problems, Acta Math. Univ. Comenianae, LXXI(1), 2002, 19-26.
• [21] V. Kondratiev, V. Liskevich, Z. Sobol, Second-order semilinear elliptic inequalities in exterior domains, J. Differential Equations, 187, 2003, 429-455.
• [22] A.A. Kon’kov, On solutions of quasi-linear elliptic inequalities containing terms with lower-order derivatives, Nonlinear Anal., 90, 2013, 121-134.
• [23] V.V. Kurta, On the nonexistence of positive solutions to semilinear elliptic equations, Proceedings of the Steklov Institute of Mathematics, 227, 1999, 155-162.
• [24] H.A. Levine, The role of critical exponents in blowup theorems, SIAM Review, 32(2), 1990, 262-288.
• [25] V. Liskevich, I.I. Skrypnik, I.V. Skrypnik, Positive super-solutions to general nonlinear elliptic equations in exterior domains, Manuscripta Math., 115, 2004, 521-538.
• [26] E. Mitidieri, S. Pohozaev, Absence of positive solutions for quasilinear elliptic problems in RN, Proc. Steklov Inst. Math., 227, 1999, 186-216 (in Russian).
• [27] E. Mitidieri, S.Z. Pohozhayev, A priori estimations and no solutions of nonlinear partial equations and inequalities, Proc. of V.A. Steklov Mathematics Institute of NAS, 234, 2001, 1-383.
• [28] R.G. Pinsky, Existence and nonexistence of global solutions for ut = ∆u +a(x) up in Rd, Journal of Differential Equations,133, 1997, 152-177.
• [29] P. Quittner, Multiple equilibria, periodic solutions and a priori bounds for solutions in superlinear parabolic problems, Nonlinear Differential Equations and Applications, 11(2), 2004, 237-258.
• [30] A.A. Samarskii, V.A. Galaktionov, S.P. Kurdyumov, A.P. Mikhaylov, Blow up of solutions in problems for quasilinear parabolic equations, Nauka, Moscow, 1987 (in Russian).
• [31] T.I. Seidman, Periodic solutions of a non-linear parabolic equation, Journal of Differential Equations, 19(2), 1975, 242-257.