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We establish the local (in time) solvability in the classical sense for the Cauchy problem and first and second boundary‐value problems on the half‐line for a nonlinear equation similar to Benjamin‐Bona‐Mahony‐Bürgers‐type equation. We also derive an a priori estimate that implies sufficient blow‐up conditions for the second boundary‐value problem. We obtain analytically an upper bound of the blow‐up time and refine it numerically using Richardson effective accuracy order technique. [ABSTRACT FROM AUTHOR]

This work develops the theory of the blow‐up phenomena for Joseph–Egri equation. The existence of the nonextendable solution of two initial‐boundary value problems (on a segment and a half‐line) is demonstrated. Sufficient conditions of the finite‐time blow‐up of these solutions, as well as the analytical estimates of the blow‐up time, are obtained. A numerical method that allows to precise the blow‐up moment for specified initial data is proposed. [ABSTRACT FROM AUTHOR]

The initial-boundary value problems for a Sobolev equation with exponential nonlinearities, classical, and nonclassical boundary conditions are considered. For this model, which describes processes in crystalline semiconductors, the blow-up phenomena are studied. The sufficient blow-up conditions and the blow-up time are analyzed by the method of the test functions. This analytical a priori information is used in the numerical experiments, which are able to determine the process of the solution's blow-up more accurately. The model derivation and some questions of local solvability and uniqueness are also discussed. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]