Abstract:
In this paper, we study the global solvability of well-known equations used to describe nonlinear processes with dissipation, namely, the Burgers equation, the Korteweg–de Vries–Burgers equation, and the modified Korteweg–de Vries–Burgers equation. Using a method due to Pokhozhaev, we obtain necessary conditions for the blow-up of global solutions and estimates of the blow-up time and blow-up rate in bounded and unbounded domains. We also study the effect of linear and nonlinear viscosity on the occurrence of a gradient catastrophe in finite time.
Keywords:
Burgers equation, global unsolvability of Burgers-type equations, Korteweg–de Vries–Burgers equation, nonlinear process with dissipation, blow-up time, gradient catastrophe, maximum principle, method of test functions.
Citation:
E. V. Yushkov, M. O. Korpusov, “Global Unsolvability of One-Dimensional Problems for Burgers-Type Equations”, Mat. Zametki, 98:3 (2015), 448–462; Math. Notes, 98:3 (2015), 503–514
This publication is cited in the following 6 articles:
Alsaedi A. Ahmad B. Kirane M. Torebek B.T., “Blowing-Up Solutions of the Time-Fractional Dispersive Equations”, Adv. Nonlinear Anal., 10:1 (2021), 952–971
Alotaibi M., Jleli M., Samet B., “Blow-Up of Solutions to Fractional-in-Space Burgers-Type Equations”, Fractal Fract., 5:4 (2021), 249
A. Alsaedi, M. Kirane, B. T. Torebek, “Blow-up of smooth solutions of the time-fractional Burgers equation”, Quaest. Math., 43:2 (2020), 185–192
B. T. Torebek, “Global unsolvability of the Burgers equation with fractional time derivative”, Differ. Equ., 55:6 (2019), 867–870
M. O. Korpusov, E. V. Yushkov, “Global unsolvability of a nonlinear conductor model in the quasistationary approximation”, Theoret. and Math. Phys., 191:1 (2017), 471–479
E. V. Yushkov, M. O. Korpusov, “Gradient blow-up in generalized Burgers and Boussinesq equations”, Izv. Math., 81:6 (2017), 1286–1296
Citation in format AMSBIB
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