## Diffusion-induced blowup solutions for the shadow limit model of a singular Gierer-Meinhardt system.(English)Zbl 1477.35043

### MSC:

 35B44 Blow-up in context of PDEs 35B40 Asymptotic behavior of solutions to PDEs 35K20 Initial-boundary value problems for second-order parabolic equations 35K58 Semilinear parabolic equations 35R09 Integro-partial differential equations

### Citations:

Zbl 1372.35138; Zbl 1387.00004; Zbl 0910.35020; Zbl 1425.35116
Full Text:

### References:

 [1] Bobrowski, A. and Kunze, M., Irregular convergence of mild solutions of semilinear equations, J. Math. Anal. Appl.472 (2019) 1401-1419. · Zbl 1411.34085 [2] Bressan, A., Stable blow-up patterns, J. Differential Equations98 (1992) 57-75. · Zbl 0770.35010 [3] Bricmont, J. and Kupiainen, A., Universality in blow-up for nonlinear heat equations, Nonlinearity7 (1994) 539-575. · Zbl 0857.35018 [4] O. Drosinou, N. I. Kavallaris and C. V. Nikolopoulos, A study of a non-local problem with robin boundary conditions arising from MEMS technology, to appear in Math. Methods Appl. Sci. (2021), https://doi.org/10.1002/mma.7393. · Zbl 1473.35340 [5] Duong, G. K., A blowup solution of a complex semi-linear heat equation with an irrational power, J. Differential Equations267 (2019) 4975-5048. · Zbl 1428.35143 [6] Duong, G. K., Profile for the imaginary part of a blowup solution for a complex-valued semilinear heat equation, J. Funct. Anal.277 (2019) 1531-1579. · Zbl 1417.35064 [7] Duong, G. K., Nguyen, V. T. and Zaag, H., Construction of a stable blowup solution with a prescribed behavior for a non-scaling-invariant semilinear heat equation, Tunis. J. Math.1 (2019) 13-45. · Zbl 1407.35092 [8] G. K. Duong, N. Nouaili and H. Zaag, Construction of blow-up solutions for the complex Ginzburg-Landau equation with critical parameters, to appear in Mem. Amer. Math. Soc. (2021). [9] Duong, G. K. and Zaag, H., Profile of a touch-down solution to a nonlocal MEMS model, Math. Models Methods Appl. Sci.29 (2019) 1279-1348. · Zbl 1425.35116 [10] Ghoul, T., Nguyen, V. T. and Zaag, H., Construction and stability of blowup solutions for a non-variational semilinear parabolic system, Ann. I. H. Poincaré AN.35(6) (2018) 1577-1630. · Zbl 1394.35222 [11] Gierer, A. and Meinhardt, H. A., Theory of biological pattern formation, Kybernetik.12 (1972) 30-39. · Zbl 1434.92013 [12] Giga, Y. and Kohn, R. V., Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math.38 (1985) 297-319. · Zbl 0585.35051 [13] Giga, Y. and Kohn, R. V., Nondegeneracy of blowup for semilinear heat equations, Comm. Pure Appl. Math.42 (1989) 845-884. · Zbl 0703.35020 [14] Guo, J. and Souplet, P., No touchdown at zero points of the permittivity profile for the MEMS problem, SIAM J. Math. Anal.47 (2015) 614-625. · Zbl 1332.35160 [15] Guo, J. S. and Kavallaris, N. I., On a nonlocal parabolic problem arising in electrostatic MEMS control, Discrete Contin. Dyn. Syst.32 (2012) 1723-1746. · Zbl 1243.35009 [16] Kavallaris, N. I., Barreira, R. and Madzvamuse, A., Dynamics of shadow system of a singular gierer-meinhardt system on an evolving domain, J. Nonlinear Sci.31(5) (2021), https://doi.org/10.1007/s00332-020-09664-3. · Zbl 1462.35033 [17] Kavallaris, N. I., Lacey, A. A. and Nikolopoulos, C. V., On the quenching of a nonlocal parabolic problem arising in electrostatic MEMS control, Nonlinear Anal.138 (2016) 189-206. · Zbl 1334.35340 [18] N. I. Kavallaris and E. A. Latos, Diffusion-driven blow-up for a non-local fisher-kpp type model. Submitted (2019). [19] Kavallaris, N. I. and Suzuki, T., On the dynamics of a non-local parabolic equation arising from the Gierer-Meinhardt system, Nonlinearity30(5) (2017) 1734-1761. · Zbl 1372.35138 [20] Kavallaris, N. I. and Suzuki, T., Non-Local Partial Differential Equations for Engineering and Biology: Mathematical Modeling and Analysis, , Vol. 31 (Springer, 2018). · Zbl 1387.00004 [21] Mahmoudi, F., Nouaili, N. and Zaag, H., Construction of a stable periodic solution to a semilinear heat equation with a prescribed profile, Nonlinear Anal.131 (2016) 300-324. · Zbl 1334.35145 [22] Marciniak-Czochra, A., Härting, S., Karch, G. and Suzuki, K., Dynamical spike solutions in a nonlocal model of pattern formation, Nonlinearity31 (2018) 1757-1781. · Zbl 1391.35289 [23] Marciniak-Czochra, A. and Mikelić, A., Shadow limit using renormalization group method and center manifold method, Vietnam J. Math.45 (2017) 103-125. · Zbl 1376.37116 [24] Masmoudi, N. and Zaag, H., Blow-up profile for the complex Ginzburg-Landau equation, J. Funct. Anal.255 (2008) 1613-1666. · Zbl 1158.35016 [25] Merle, F., Solution of a nonlinear heat equation with arbitrarily given blow-up points, Comm. Pure Appl. Math.45 (1992) 263-300. · Zbl 0785.35012 [26] Merle, F. and Zaag, H., Reconnection of vortex with the boundary and finite time quenching, Nonlinearity10 (1997) 1497-1550. · Zbl 0910.35020 [27] Merle, F. and Zaag, H., Stability of the blow-up profile for equations of the type $$u_t=\operatorname{\Delta}u+|u |^{p - 1}u$$, Duke Math. J.86 (1997) 143-195. · Zbl 0872.35049 [28] V. T. Nguyen and H. Zaag, Construction of a stable blow-up solution for a class of strongly perturbed semilinear heat equations, to appear in Ann. Scuola Norm. Sup. Pisa Cl. Sci. · Zbl 1378.35178 [29] Nouaili, N. and Zaag, H., Profile for a simultaneously blowing up solution to a complex valued semilinear heat equation, Comm. Partial Differential Equations40 (2015) 1197-1217. · Zbl 1335.35126 [30] Nouaili, N. and Zaag, H., Construction of a blow-up solution for the complex Ginzburg-Landau equation in some critical case, Arch. Ration. Mech. Anal.228(3) (2018) 995-1058. · Zbl 1397.35295 [31] S. Tayachi and H. Zaag, Existence of a stable blow-up profile for the nonlinear heat equation with a critical power nonlinear gradient term, to appear Trans. Amer. Math. Soc.371 (2019) 5899-5972. · Zbl 1423.35186 [32] Turing, A. M., The chemical basis of morphogenesis, Philos. Trans. R. Soc.237 (1952) 37-72. · Zbl 1403.92034 [33] Yang, X. and Zhang, T., Estimates of heat kernels with Neumann boundary conditions, Potential Anal.38 (2013) 549-572. · Zbl 1262.35115
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.