Abstract:
We consider two different abstract Cauchy problems for equations of Sobolev type with operator coefficients in Banach spaces. For the first problem we obtain, under certain conditions on the coefficients, optimal theorems on the existence and non-existence of a solution global in time. In the case when the solution is blown up we obtain upper and lower bounds for the blow-up time. For the second problem we obtain optimal upper and lower bounds for the rate of blow-up of a solution. In each case we give examples in which the operator coefficients have a physical meaning.
\Bibitem{Kor04}
\by M.~O.~Korpusov
\paper Blow-up of solutions of a~class of strongly non-linear equations of Sobolev type
\jour Izv. Math.
\yr 2004
\vol 68
\issue 4
\pages 783--832
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\crossref{https://doi.org/10.1070/IM2004v068n04ABEH000498}
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https://www.mathnet.ru/eng/im498
https://doi.org/10.1070/IM2004v068n04ABEH000498
https://www.mathnet.ru/eng/im/v68/i4/p151
This publication is cited in the following 10 articles:
Zhang H. Hu Q. Liu G., “Global Existence, Asymptotic Stability and Blow-Up of Solutions For the Generalized Boussinesq Equation With Nonlinear Boundary Condition”, Math. Nachr., 293:2 (2020), 386–404
M. O. Korpusov, “Blow-up and global solubility in the classical sense of the Cauchy problem for a formally hyperbolic equation with a non-coercive source”, Izv. Math., 84:5 (2020), 930–959
Piskin E., Ekinci F., “Blow Up, Exponential Growth of Solution For a Reaction-Diffusion Equation With Multiple Nonlinearities”, Tbil. Math. J., 12:4 (2019), 61–70
Antontsev S.N., de Oliveira H.B., Khompysh Kh., “Generalized Kelvin-Voigt Equations For Nonhomogeneous and Incompressible Fluids”, Commun. Math. Sci., 17:7 (2019), 1915–1948
M. O. Korpusov, “Blow-up of solutions of nonclassical nonlocal nonlinear model equations”, Comput. Math. Math. Phys., 59:4 (2019), 583–609
Erhan Pişkin, Fatma Ekinci, “Blow up, exponential growth of solution for a reaction-diffusion equation with multiple nonlinearities”, Tbilisi Math. J., 12:4 (2019)
Hongwei Zhang, Jun Lu, Qingying Hu, “Exponential growth of solution of a strongly nonlinear generalized Boussinesq equation”, Computers & Mathematics with Applications, 2014
Blow-up in Nonlinear Sobolev Type Equations, 2011, 621
M. O. Korpusov, A. G. Sveshnikov, “Blow-up of solutions of a class of strongly non-linear dissipative wave
equations of Sobolev type with sources”, Izv. Math., 69:4 (2005), 733–770
M. O. Korpusov, A. G. Sveshnikov, “On blowup of a solution to a Sobolev-type equation with a nonlocal source”, Siberian Math. J., 46:3 (2005), 443–452
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