A necessary and sufficient condition for the existence of global solutions to reaction-diffusion equations on bounded domains

Springer Science and Business Media LLC
Soon–Yeong Chung1, Jaeho Hwang2
1Department of Mathematics and Program of Integrated Biotechnology, Sogang University, Seoul, Republic of Korea
2Research Institute for Basic Science, Sogang University, Seoul, Republic of Korea

Tóm tắt

AbstractThe purpose of this paper is to give a necessary and sufficient condition for the existence and non-existence of global solutions of the following semilinear parabolic equations $$ u_{t}=\Delta u+\psi (t)f(u),\quad \text{in }\Omega \times (0,\infty ), $$ u t = Δ u + ψ ( t ) f ( u ) , in  Ω × ( 0 , ∞ ) , under the mixed boundary condition on a bounded domain Ω. In fact, this has remained an open problem for a few decades, even for the case $f(u)=u^{p}$ f ( u ) = u p . As a matter of fact, we prove: $$ \begin{aligned} & \text{there is no global solution for any initial data if and only if } \\ & \int _{0}^{\infty}\psi (t) \frac{f (\lVert S(t)u_{0}\rVert _{\infty} )}{\lVert S(t)u_{0}\rVert _{\infty}}\,dt= \infty \\ &\text{for every nonnegative nontrivial initial data } u_{0}\in C_{0}( \Omega ). \end{aligned} $$ there is no global solution for any initial data if and only if  ∫ 0 ∞ ψ ( t ) f ( ∥ S ( t ) u 0 ∥ ∞ ) ∥ S ( t ) u 0 ∥ ∞ d t = ∞ for every nonnegative nontrivial initial data  u 0 ∈ C 0 ( Ω ) . Here, $(S(t))_{t\geq 0}$ ( S ( t ) ) t ≥ 0 is the heat semigroup with the mixed boundary condition.

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Tài liệu tham khảo

Fujita, H.: On the blowing up of solutions of the Cauchy problems for $u_{t}=\delta u+u^{1+\alpha}$. J. Fac. Sci., Univ. Tokyo, Sect. 1 13, 109–124 (1966)

Hayakawa, K.: On nonexistence of global solutions of some semilinear parabolic differential equations. Proc. Jpn. Acad. 49, 503–505 (1973)

Kobayashi, K., Siaro, T., Tanaka, H.: On the blow-up problem for semilinear heat equations. J. Math. Soc. Jpn. 29, 407–424 (1977)

Deng, K., Levine, H.A.: The role of critical exponents in blow-up theorems: the sequel. J. Math. Anal. Appl. 243, 85–126 (2000)

Levine, H.A.: The role of critical exponents in blowup theorems. SIAM Rev. 32(2), 262–288 (1990)

Meier, P.: On the critical exponent for reaction-diffusion equations. Arch. Ration. Mech. Anal. 109, 63–71 (1990)

Bai, X., Zheng, S., Wang, W.: Critical exponent for parabolic system with time-weighted sources in bounded domain. J. Funct. Anal. 6, 941–952 (2013)

Qi, Y.: The critical exponents of parabolic equations and blow-up in $\mathbb{R}^{N}$. Proc. R. Soc. Edinb., Sect. A 128(1), 123–136 (1998)

Qi, Y.-W., Wang, M.-X.: Critical exponents of quasilinear parabolic equations. J. Math. Anal. Appl. 1, 264–280 (2002)

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. 168, 9 (2014)

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.: Global existence and blow up for a coupled parabolic system with time-weighted sources on a general domain. Z. Angew. Math. Phys. 70, 16 (2019)

Hu, B.: Blow-up Theories for Semilinear Parabolic Equations. Lecture Notes in Mathematics, vol. 2018. Springer, Heidelberg (2011)

Chung, S.-Y., Hwang, J.: A necessary and sufficient condition for the existence of global solutions to discrete semilinear parabolic equations on networks. Chaos Solitons Fractals 158, 112055 (2022)

Cano-Casanova, S.: Properties of the principal eigenvalues of a general class of non-classical mixed boundary value problems. J. Differ. Equ. 178(1), 123–211 (2002)

Lê, A.: An eigenvalue problems for the p-Laplacian. Nonlinear Anal. 64(5), 1057–1099 (2006)

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