Geodesic
Gradient blow-up in generalized Burgers and Boussinesq equations
Izvestiya. Mathematics , Tome 81 (2017) no. 6, pp. 1286-1296

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We study the influence of gradient non-linearity on the global solubility of initial-boundary value problems for a generalized Burgers equation and an improved Boussinesq equation which are used for describing one-dimensional wave processes in dissipative and dispersive media. For a large class of initial data, we obtain sufficient conditions for global insolubility and a bound for blow-up times. Using the Boussinesq equation as an example, we suggest a modification of the method of non-linear capacity which is convenient from a practical point of view and enables us to estimate the blow-up rate. We use the method of contraction mappings to study the possibility of instantaneous blow-up and short-time existence of solutions.
Keywords: gradient non-linearity, Burgers equation and generalized Boussinesq equations, blow-up phenomena, method of non-linear capacity. 
@article{IM2_2017_81_6_a10,
     author = {E. V. Yushkov and M. O. Korpusov},
     title = {Gradient blow-up in generalized {Burgers} and {Boussinesq} equations},
     journal = {Izvestiya. Mathematics },
     pages = {1286--1296},
     publisher = {mathdoc},
     volume = {81},
     number = {6},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2017_81_6_a10/}
}
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E. V. Yushkov; M. O. Korpusov. Gradient blow-up in generalized Burgers and Boussinesq equations. Izvestiya. Mathematics , Tome 81 (2017) no. 6, pp. 1286-1296. http://geodesic.mathdoc.fr/item/IM2_2017_81_6_a10/