On blow-up set of solutions of initial boundary value problem for a system of parabolic equations with nonlocal boundary conditions

A. Gladkov ; A. I. Nikitin

Journal of the Belarusian State University. Mathematics and Informatics, Tome 2 (2018), pp. 17-24.

Résumé

We consider a system of semilinear parabolic equations $u_{t}=\Delta u + c_{1}(x,t)\nu^{p}, \nu_{t}=\Delta\nu+c_{2}(x,t)u^{q}, (x,t)\in \Omega\times (0,+\infty)$ with nonlinear nonlocal boundary conditions $\dfrac{\partial u}{\partial\eta}= \int\limits_\Omega k_{1}(x,y,t)u^{m}(y,t)dy, \dfrac{\partial\nu}{\partial\eta}= \int\limits_\Omega k_{2}(x,y,t)\nu^{n}(y,t)dy, (x,t)\in \partial\Omega\times (0,+\infty)$ and initial data $u(x,0)=u_{0}(x), \nu(x,0)=\nu_{0}(x), x\in \Omega$, where $p,q,m,n$ are positive constants, $\Omega$ is bounded domain in $\mathbb{R}^{N}(N\geq 1)$ with a smooth boundary $\partial\Omega, \eta$ is unit outward normal on $\partial\Omega$. Nonnegative locally Holder continuous functions $c_{i}(x,t), i=1,2$, are defined for $x\in \overline{\Omega}, t\geq 0$; nonnegative continuous functions $k_{i}(x,y,t), i=1,2$ are defined for $x\in \partial\Omega, y\in \overline{\Omega}, t\geq 0$; nonnegative continuous functions $u_{0}(x), \nu_{0}(x)$ are defined for $x\in \overline{\Omega}$ and satisfy the conditions $\dfrac{\partial u_{0}(x)}{\partial\eta}= \int\limits_\Omega k_{1}(x,y,0)u_{0}^{m}(y)dy, \dfrac{\partial \nu_{0}(x)}{\partial\eta}= \int\limits_\Omega k_{2}(x,y,0)\nu_{0}^{n}(y)dy$ for $x\in \partial\Omega$. In the paper blow-up set of classical solutions is investigated. It is established that blow-up of the solutions can occur only on the boundary $\partial\Omega$ if $max(p,q)\leq 1, max(m,n)>1$ and under certain conditions for the coefficients $k_{i}(x,y,t), i=1,2$.

Détail

Keywords: system of semilinear parabolic equations, nonlocal boundary conditions, blow-up set.

Bibliographie

[1] Z. Cui, Z. Yang, “Roles of weight functions to a nonlinear porous medium equation with nonlocal source and nonlocal boundary condition”, Journal of Mathematical Analysis Applications, 342(1) (2008), 559–570 | DOI | MR | Zbl

[2] Z. Cui, Z. Yang, R. Zhang, “Blow-up of solutions for nonlinear parabolic equation with nonlocal source and nonlocal boundary condition”, Applied of Mathematics and Computation, 224 (2013), 1–8 | DOI | MR | Zbl

[3] K. Deng, Z. Dong, “Blow-up for the equation with a general memory boundary condition”, Communications on Pure and Applied Analysis, 11 (2012), 2147–2156 | DOI | MR | Zbl

[4] Z. B. Fang, J. Zhang, “Global existence and blow-up of solutions for p-Laplacian evolution equation with nonlinear memory term and nonlocal boundary condition”, Boundary Value Problem, 2014(1) (2014), 8 | DOI | MR

[5] Y. Gao, W. Gao, “Existence and blow-up of solutions for a porous medium equation with nonlocal boundary condition”, Applicable Analysis, 90(5) (2011), 799–809 | DOI | MR | Zbl

[6] A. Gladkov, M. Guedda, “Blow-up problem for semilinear heat equation with absorption and a nonlocal boundary condition”, Nonlinear Analysis, 74(13) (2011), 4573–4580 | DOI | MR | Zbl

[7] A. Gladkov, K. I. Kim, “Blow-up of solutions for semilinear heat equation with nonlinear nonlocal boundary condition”, Journal of Mathematical Analysis and Applications, 338(1) (2008), 264–273 | DOI | MR | Zbl

[8] C. V. Pao, “Asimptotic behavior of solutions of reaction-diffusion equations with nonlocal boundary conditions”, Journal of Mathematical Analysis and Applications, 88(1) (1998), 225–238 | DOI | MR | Zbl

[9] Y. Wang, C. Mu, Z. Xiang, “Blowup of solutions to a porous medium equation with nonlocal boundary condition”, Applied of Mathematics and Computation, 192(2) (2007), 579–585 | DOI | MR | Zbl

[10] L. Yang, C. Fan, “Global existence and blow-up of solutions to a degenerate parabolic system with nonlocal sources and nonlocal boundaries”, Monatshefte fur Mathematik, 174 (2014), 493–510 | DOI | MR | Zbl

[11] Z. Ye, X. Xu, “Global existence and blow-up for a porous medium system with nonlocal boundary conditions and nonlocal sources”, Nonlinear Analysis: Theory, Methods and Applications, 82 (2013), 115–126 | DOI | MR | Zbl

[12] H. M. Yin, “On a class of parabolic equations with nonlocal boundary conditions”, Journal of Mathematical Analysis and Applications, 294(2) (2004), 712–728 | DOI | MR | Zbl

[13] S. Zheng, I. Kong, “Roles of weight functions in a nonlinear nonlocal parabolic system”, Nonlinear Analysis: Theory, Methods and Applications, 68(8) (2008), 2406–2416 | DOI | MR | Zbl

[14] A. Gladkov, T. Kavitova, “Blow-up problem for semilinear heat equation with nonlinear nonlocal boundary condition”, Applicable Analysis, 95(9) (2016), 1974–1988 | DOI | MR | Zbl

[15] A. Gladkov, T. Kavitova, “Initial boundary value problem for a semilinear parabolic equation with nonlinear nonlocal boundary conditions”, Ukrainian Mathematics Journal, 68(2) (2016), 179–192 | DOI | MR | Zbl

[16] A. I. Nikitin, “Lokalnoe suschestvovanie reshenii nachalno-kraevoi zadachi dlya sistemy polulineinykh parabolicheskikh uravnenii s nelineinymi nelokalnymi granichnymi usloviyami”, Vesnik VDU, 5 (2015), 14–19

[17] A. L. Gladkov, A. I. Nikitin, “O globalnom suschestvovanii reshenii nachalno-kraevoi zadachi dlya sistemy polulineinykh parabolicheskikh uravnenii s nelineinymi nelokalnymi granichnymi usloviyami Neimana”, Differentsialnye uravneniya, 54(1) (2018), 88–107 | DOI | Zbl

[18] C. S. Kahane, “On the asymptotic behavior of solutions of parabolic equations”, Czechoslovak Mathematics Journal, 33(2) (1983), 262–285 | DOI | MR | Zbl

[19] B. Hu, H. M. Yin, “Critical exponents for a system of heat equations coupled in a non-linear boundary condition”, Mathematical Methods in the Applied Sciences, 19(14) (1996), 1099–1120 | 3.0.CO;2-J class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[20] K. Deng, C. L. Zhao, “Blow-up for a parabolic system coupled in an equation and a boundary condition”, Proceedings of the Royal Society of Edinburg Section A, 131(6) (2001), 1345–1355 | DOI | MR | Zbl

[21] B. Hu, H. M. Yin, “The profile near blowup time for solution of the heat equation with a nonlinear boundary condition”, Transactions American Mathematical Society, 346(1) (1994), 117–135 | DOI | MR | Zbl