Abstract
Two Cauchy problems for the nonlinear Sobolev equations \(\frac{{{{\partial }^{2}}}}{{\partial {{t}^{2}}}}\frac{{{{\partial }^{2}}u}}{{\partial x_{3}^{2}}} + \Delta u = {{\left| u \right|}^{q}}\) and \(\frac{{{{\partial }^{2}}}}{{\partial {{t}^{2}}}}{{\Delta }_{ \bot }}u + \Delta u = {{\left| u \right|}^{q}}\) are investigated. Conditions are found under which the Cauchy problems have weak generalized local-in-time solutions, and the blow-up conditions for weak solutions of these problems are determined.
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Korpusov, M.O., Shafir, R.S. On Cauchy Problems for Nonlinear Sobolev Equations in Ferroelectricity Theory. Comput. Math. and Math. Phys. 62, 2091–2111 (2022). https://doi.org/10.1134/S0965542522120089
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DOI: https://doi.org/10.1134/S0965542522120089