Existence and non-existence of global solutions for a heat equation with degenerate coefficients
Tóm tắt
In this paper, the parabolic problem
$$u_t - div(\omega (x) \nabla u)= h(t) f(u) + l(t) g(u)$$
with non-negative initial conditions pertaining to
$$C_b({\mathbb {R}}^N)$$
, will be studied, where the weight
$$\omega $$
is an appropriate function that belongs to the Muckenhoupt class
$$A_{1 + \frac{2}{N}}$$
and the functions f, g, h and l are non-negative and continuous. The main goal is to establish the global and non-global existence of non-negative solutions. In addition, will be obtained both the so-called Fujita’s exponent and the second critical exponent in the sense of Lee and Ni (Trans Am Math Soc 333(1):365–378, 1992), in the particular case when
$$h(t)\sim t^r \,(r>-1)$$
,
$$l(t)\sim t^s \, (s>-1)$$
,
$$f(u)=u^p$$
and
$$g(u)=(1+u)[\ln (1+u)]^p$$
. The results of this paper extend those obtained by Fujishima et al. (Calc Var Partial Differ Equ 58:62, 2019) that worked when
$$h(t)=1$$
,
$$l(t)=0$$
and
$$f(u)=u^p $$
.
Tài liệu tham khảo
Ahmad, B., Alsaedi, A., Berbiche, M., Kirane, M.: Existence of global solutions and blow-up of solutions for coupled systems of fractional diffusion equations. Electron. J. Differ. Equ. 2020, 1–28 (2020)
Aronson, D.G., Weinberger, H.F.: Multidimensional nonlinear diffusion arising in population genetics. Adv. Math. 30(1), 33–76 (1978)
Bai, X., Zheng, S., Wang, W.: Critical exponent for parabolic system with time-weighted sources in bounded domain. J. Funct. Anal. 265, 941–952 (2013)
Bonforte, M., Simonov, N.: Quantitative a priori estimates for fast diffusion equations with Caffarelli–Kohn–Nirenberg weights. Harnack inequalities and Holder continuity. Adv. Math. 345, 1075–1161 (2019)
Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32, 1245–1260 (2007)
Castillo, R., Loayza, M.: On the critical exponent for some semilinear reaction–diffusion systems on general domains. J. Math. Anal. Appl. 428, 1117–1134 (2015)
Castillo, R., Loayza, M., Paixão, C.S.: Global and nonglobal existence for a strongly coupled parabolic system on a general domain. J. Differ. Equ. 261, 3344–3365 (2016)
Castillo, R., Loayza, M.: Global existence and blowup for a coupled parabolic system with time-weighted sources on a general domain. Z. Angew. Math. Phys. 70, 16 (2019)
Castillo, R., Guzmán-Rea, O., Loayza, M., Zegarra, M.: Global solution for a coupled parabolic system with degenerate coefficients and time-weighted sources (2022). arXiv:2209.04781v1 [math.AP]
Cazenave, T., Haraux, A.: An introduction to semilinear evolution equations. Transl. by Yvan Martel. Revised ed., Oxford Lecture Series in Mathematics and its Applications, vol. 13, p. 14. Clarendon Press, Oxford (1998). ISBN 0-19-850245-X
Chen, H., Luo, P., Liu, G.: Global solution and blow up of a semilinear heat equation with logarithmic nonlinearity. J. Math. Anal. Appl. 422, 84–98 (2015)
Deng, X.M., Zhou, J.: Global existence and blow-up of solutions to a semilinear heat equation with singular potential and logarithmic nonlinearity. Commun. Pure Appl. Anal. 19(2), 923–939 (2020)
Ding, J.T.: Blow-up solutions and global solutions for a class of quasilinear parabolic equations with Robin boundary conditions. Comput. Math. Appl. 49, 689–701 (2005)
Escobedo, M., Herrero, M.A.: Boundedness and blow up for a semilinear reaction–diffusion system. J. Differ. Equ. 89, 176–202 (1991)
Ferreira, L.C.F., de Queiroz, O.S.: A singular parabolic equation with logarithmic nonlinearity \(L^{p}\)-initial data. J. Differ. Equ. 249, 349–365 (2010)
Ferreira, R., de Pablo, A., Rossi, J.D.: Blow-up with logarithmic nonlinearities. J. Differ. Equ. 240, 196–215 (2007)
Fujishima, Y., Kawakami, T., Sire, Y.: Critical exponent for the global existence of solutions to a semilinear heat equation with degenerate coefficients. Calc. Var. Partial Differ. Equ. 58, 62 (2019)
Fujita, H.: On the blowing up of solutions of the Cauchy problem for \(u_t=\Delta u+u^{1+\alpha }\). J. Fac. Sci. Univ. Tokyo Sect. I(13), 109–124 (1966)
Galaktionov, V.A., Kurdyumov, S.P., Mikhailov, A.P.: Unbounded solutions of semilinear parabolic equations. Keldysh Inst. Appl. Math. Acad. Sci. USSR, Preprint, No. 161 (1979)
Galaktionov, V.A.: On global unsolvability of Cauchy problems for quasilinear parabolic equations. Zh. Vychisl. Matem. Matem. Fiz. 23, 1072–1087 (1983) (in Russian); English translation: USSR Comput. Math. Math. Phys. 23, 31–41 (1983)
Galaktionov, V.A., Vazquez, J.L.: Blow-up for quasilinear heat equations described by means of nonlinear Hamilton–Jacobi equations. J. Differ. Equ. 27, 1–40 (1996)
Galaktionov, V.A., Vazquez, J.L.: Regional blow-up in a semilinear heat equation with convergence to a Hamilton–Jacobi equation. SIAM J. Math. Anal. 24, 1254–1276 (1993)
Galaktionov, V.A., Levine, H.A.: A general approach to critical Fujita exponents in nonlinear parabolic problems. Nonlinear Anal. 34, 1005–1027 (1998)
Gutiérrez, C.E., Nelson, G.S.: Bounds for the fundamental solution of degenerate parabolic equations. Comm. Partial Differ. Equ. 13, 635–649 (1988)
Gutiérrez, C.E., Wheeden, R.L.: Bounds for the fundamental solution of degenerate parabolic equations. Comm. Partial Differ. Equ. 17(7–8), 1287–1307 (1992)
Gutierrez, C.E.: Pointwise estimates for solutions of degenerate parabolic equations. Rev. U. Mat. Argent. 37, 261–270 (1991)
Hayakawa, K.: On nonexistence of global solutions of some semilinear parabolic equations. Proc. Jpn. Acad. 49, 503–505 (1973)
Kamin, S., Rosenau, P.: Propagation of thermal waves in an inhomogeneous medium. Commun. Pure Appl. Math. 34, 831–852 (1981)
Kamin, S., Rosenau, P.: Nonlinear thermal evolution in an inhomogeneous medium. J. Math. Phys. 23, 1385–1390 (1982)
Klafter, J., Sokolov, I.M.: Anomalous diffusion spreads its wings. Phys. World 18, 29–32 (2005)
Kobayashi, K., Sirao, T., Tanaka, H.: On blowing up problem for semilinear heat equation. J. Math. Soc. Jpn. 29, 407–424 (1977)
Laister, R., Robinson, J.C., Sierzega, M.: Non-existence of local solutions for semilinear heat equations of Osgood type. J. Differ. Equ. 255, 3020–3028 (2013)
Lee, T.Y., Ni, W.M.: Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem. Trans. Am. Math. Soc. 333(1), 365–378 (1992)
Lian, W., Ahmed, M.S., Xu, R.: Global existence and blow up of solution for semilinear hyperbolic equation with logarithmic nonlinearity. Nonlinear Anal. 184, 239–257 (2019)
Li, Z., Mu, C.: Critical exponents and blow-up rate for a nonlinear diffusion equation with logarithmic boundary flux. Nonlinear Anal. 73, 933–939 (2010)
Loayza, M., da Paixão, C.S.: Existence and non-existence of global solutions for a semilinear heat equation on a general domain. Electron. J. Differ. Equ. 2014, 1–9 (2014)
Meier, P.: Blow-up of solutions of semilinear parabolic differential equations. J. Appl. Math. Phys. (ZAMP) 39, 135–149 (1988)
Nyström, K., Sande, O.: Extension properties and boundary estimates for a fractional heat operator. Nonlinear Anal. 140, 29–37 (2016)
Quittner, P., Souplet, P.: Superlinear Parabolic Problems: Blow-Up, Global Existence and Steady States. Springer, Berlin (2007)
Rosenau, P., Kamin, S.: Nonlinear diffusion in finite mass medium. Commun. Pure Appl. Math. 35, 113–127 (1982)
Weissler, F.B.: Existence and non-existence of global solutions for a semilinear heat equation. Isr. J. Math. 38, 29–40 (1981)
Yang, C.X., Cao, Y., Zheng, S.N.: Life span and second critical exponent for semilinear pseudo-parabolic equation. J. Differ. Equ. 253, 3286–3303 (2012)
Yang, J.: Second critical exponent for a nonlinear nonlocal diffusion equation. Appl. Math. Lett. 81, 57–62 (2018)
Yang, J., Yang, C., Zheng, S.: Second critical exponent for evolution \(p\)-Laplacian equation with weighted source. Math. Comput. Model. 56, 247–256 (2012)
Zeldovich, Y., Raizer, Y.: Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena. Academic Press, New York (1966)