Geodesic
Weighted Identities for the Solutions of Generalized Korteweg--de Vries Equations
Matematičeskie zametki, Tome 89 (2011) no. 3, pp. 393-409

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Consider the Korteweg–de Vries equation $u_t+u_{xxx}+uu_{x}=0$ and its generalization $u_t+u_{xxx}+f(u)_{x}=0$. For the solutions of these equations, weighted identities (differential and integral) are obtained. These identities make it possible to establish the blow-up (in finite time) of the solutions of certain boundary-value problems.
Keywords: Korteweg–de Vries equation, initial boundary-value problem, weighted differential inequality, weighted integral inequality, blow-up of solutions, Hölder's inequality, Young's inequality, Dirichlet boundary condition. 
@article{MZM_2011_89_3_a8,
     author = {S. I. Pokhozhaev},
     title = {Weighted {Identities} for the {Solutions} of {Generalized} {Korteweg--de} {Vries} {Equations}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {393--409},
     publisher = {mathdoc},
     volume = {89},
     number = {3},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2011_89_3_a8/}
}
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S. I. Pokhozhaev. Weighted Identities for the Solutions of Generalized Korteweg--de Vries Equations. Matematičeskie zametki, Tome 89 (2011) no. 3, pp. 393-409. http://geodesic.mathdoc.fr/item/MZM_2011_89_3_a8/