Propiedades del soporte de soluciones de una clase de ecuaciones de evolución no lineales en dos dimensiones.

• Autores: Eddye Bustamante, José Jiménez Urrea
• Localización: Integración: Temas de matemáticas, ISSN 0120-419X, Vol. 39, Nº. 1, 2021, págs. 41-50
• Idioma: español
• DOI: 10.18273/revint.v39n1-2021003
• Enlaces
  • Texto completo (pdf)
• Resumen
  • español

    En este trabajo consideramos ecuaciones de la forma ∂tu + P(D)u + u^{t}∂xu = 0, donde P(D) es un operador diferencial en dos dimensiones, y l ∈ N. Probamos que si $u$ es una solución suficientemente suave de la ecuación, tal que u(0), supp u(T) ⊂ [−B, B] × [−B, B] para algún B>0, entonces existe R_0 > 0 tal que supp u(t) ⊂ [-R_0,R_0]×[-R_0,R_0] para todo t ∈ [0, T].

  • English

    In this work we consider equations of the form∂tu+P(D)u+ul∂xu= 0,whereP(D)is a two-dimensional differential operator, andl2N. Weprove that ifuis a sufficiently smooth solution of the equation, suchthatsuppu(0),suppu(T)[B, B][B, B]for someB >0, then thereexistsR0>0such thatsuppu(t)[R0, R0][R0, R0]for everyt2[0, T].

• Referencias bibliográficas
  • Referencias Ablowitz M., Nonlinear dispersive waves: asymptotic analysis and solitons, Cambridge University Press, 2011.
  • Benzekry S., et. al., “Classical Mathematical Models for Description and Prediction of Experimental Tumor Growth”, PLOS Comput. Biol., 10...
  • Biagioni H.A. and Linares F., “Well-posedness results for the modified Zakharov-Kuznetsov equation”, Birkhäuser, Basel, 54 (2003), 181-189.
  • Bourgain J., “On the compactness of the support of solutions of dispersive equations”, Internat. Math. Res. Notices, (1997), No. 9, 437-447....
  • Bustamante E., Isaza P. and Mejía J., “On uniqueness properties of solutions of the Zakharov–Kuznetsov equation”, J. Funct. Anal., 254 (2013),...
  • Bustamante E., Isaza P. and Mejía J., “On the support of solutions to de ZakharovKuznetsov equation”, J. Differential Equations, 251 (2011),...
  • Faminskii A.V., “The Cauchy problem for the Zakharov-Kuznetsov equation”, Differential Equations, 31 (1995), No. 6, 1002-1012.
  • Hall E.J. and Giaccia A.J., Radiobiology for the Radiologists, Lippincott Williams & Wilkins (LWW), 8th ed., Philadelphia, 2018.
  • Kenig C., Ponce G. and Vega L., “On the support of solutions to the generalized KdV equation”, Ann. Inst. H. Poincaré Anal. Non Linéaire,...
  • Larkin N.A., Kawahara-Burgers equation on a strip, Adv. Math. Phys., Maringá, (2015). doi: 10.1155/2015/269536
  • Larkin N.A., “The 2D Kawahara equation on a half-strip”, Appl. Math. Optim., 70 (2014), No. 3, 443-468. doi: 10.1007/s00245-014-9246-4
  • Linares F. and Pastor A., “Local and global well-posedness for the 2D generalizes Zakharov-Kuztnesov equation”, J. Funct. Anal., 260 (2011),...
  • Linares F. and Pastor A., “Well-posedness for the two-dimensional modified ZakharovKuznetsov equation”, SIAM J. Math. Anal., 41 (2009), No....
  • Linares F., Pastor A. and Saut J.C., “Well-Posedness for the ZK Equation in a Cylinder and on the Background of a KdV Soliton”, Comm. Partial...
  • Nahas J. and Ponce G., “On the persistent properties of solutions to semi-linear Schrödinger equation”, Comm. Partial Differential Equations,...
  • Panthee M., “A note on the unique continuation property for Zakharov-Kuznetsov equation”, Nonlinear Anal., 59 (2004), No. 3, 425-438. doi:...
  • Tao T., Nonlinear dispersive equations, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical...
  • Zakharov V.E. and Kuznetsov E.A., “On three-dimensional solitons”, Soviet Phys. JETP., 29 (1974), 594-597.