Geodesic
On the Blow-up of Solutions to Nonlinear Initial--Boundary Value Problems
Informatics and Automation, Function theory and nonlinear partial differential equations, Tome 260 (2008), pp. 213-226

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We consider the problem of nonexistence (blow-up) of solutions of nonlinear evolution equations in the case of a bounded (with respect to the space variables) domain. Following the method of nonlinear capacity based on the application of test functions that are optimal (“characteristic”) for the corresponding nonlinear operators, we obtain conditions for the blow-up of solutions to nonlinear initial–boundary value problems. We also show by examples that these conditions are sharp in the class of problems under consideration. 
@article{TRSPY_2008_260_a14,
     author = {S. I. Pokhozhaev},
     title = {On the {Blow-up} of {Solutions} to {Nonlinear} {Initial--Boundary} {Value} {Problems}},
     journal = {Informatics and Automation},
     pages = {213--226},
     publisher = {mathdoc},
     volume = {260},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2008_260_a14/}
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S. I. Pokhozhaev. On the Blow-up of Solutions to Nonlinear Initial--Boundary Value Problems. Informatics and Automation, Function theory and nonlinear partial differential equations, Tome 260 (2008), pp. 213-226. http://geodesic.mathdoc.fr/item/TRSPY_2008_260_a14/