DOI QR코드

DOI QR Code

LIOUVILLE THEOREMS OF SLOW DIFFUSION DIFFERENTIAL INEQUALITIES WITH VARIABLE COEFFICIENTS IN CONE

  • Fang, Zhong Bo (SCHOOL OF MATHEMATICAL SCIENCES, OCEAN UNIVERSITY OF CHINA) ;
  • Fu, Chao (SCHOOL OF MATHEMATICAL SCIENCES, OCEAN UNIVERSITY OF CHINA) ;
  • Zhang, Linjie (SCHOOL OF MATHEMATICAL SCIENCES, OCEAN UNIVERSITY OF CHINA)
  • 투고 : 2011.03.16
  • 심사 : 2011.03.18
  • 발행 : 2011.03.25

초록

We here investigate the Liouville type theorems of slow diffusion differential inequality and its coupled system with variable coefficients in cone. First, we give the definition of global weak solution, and then we establish the universal estimate (does not depend on the initial value) of solution by constructing test function. At last, we obtain the nonexistence of non-negative non-trivial global weak solution within the appropriate critical exponent. The main feature of this method is that we need not use comparison theorem or the maximum principle.

키워드

참고문헌

  1. C. BANDLE, H.A. LEVINE, On the existence and nonexistence of global solutions of reaction-diffusion equations in sectorial domains, Trans. Amer. Math. Soc, 316(1989), 595-622 https://doi.org/10.1090/S0002-9947-1989-0937878-9
  2. H. BERESTYCKI, I.CAPUZZO DOLCETTA AND L. NIRENBERG, Superlinear indenite elliptic problems and nonlinear Liouville theorems, Topol. Methods Nonlinear Anal, 4(1994), no.1, 59-78. https://doi.org/10.12775/TMNA.1994.023
  3. I. BIRINDELLI AND F. DEMENGEL, Some Liouville theorems for the p-Laplacian Elec. J. Differential Equations, Conference 08, 2002, 35-46.
  4. I. BIRINDELLI AND E. MITIDIERI, Liouville theorems for elliptic inequalities and applications, Proc. Royal Soc. Edinburgh Vol. 128A(1998), 1217-1247.
  5. A. BONFIGLIOLI AND E. LANCONELLI, Liouville-type theorems for real sub-Laplacians, Manuscripta Math, 105 (2001), 111-124. https://doi.org/10.1007/PL00005872
  6. G. CARISTI, Existence and nonexistence of global solutions of degenerate and singular parabolic systems, Abstr. Appl. Anal, Volume 5, Number 4 (2000), 265-284. https://doi.org/10.1155/S1085337501000380
  7. LUCIO DAMASCELLI AND FRANCESCA GLADIALI, Some nonexistence results for positive solutions of elliptic equations in unbounded domains, Rev. Mat. Iberoamericana, 20 (2004), no.1, 67-86.
  8. E.N. DANCER AND Y. DU, Some remarks on liouville type results for quasi- linear elliptic equations, Proc. Amer. Math. Soc, 131 (2002), 1891-1899.
  9. H. FUJITA, On the blowing up of solutions to the Cauchy problem for ut = $\triangle$u + u$^{1+\alpha}$, J. Fac. Sci. Univ. Tokyo, Sect. 1A. Math, 13(1966), 119-124.
  10. K. HAYAKAWA, On nonexistence of global solution of some semilinear parabolic differential equations, Proc. Japan. Acad. Ser. A, 49(1979), 503-505.
  11. A.G. KARTSATOS, V.V. KURTA, On a Liouville-type theorem and the Fujita blow-up phenomenon, Proc. Amer. Math. Soc, 132(2003), 807-813.
  12. V.A. KONDRAT'EV, Boundary-value problems for elliptic equations in domains with conic and angular points, Trans. Moscow Math. Soc, 16(1967), 209-292.
  13. G.G. LAPTEV, Absence of solutions to semilinear parabolic differential inequalities in cones, Mat. Sb, 192(2001), no.10, 51-70. https://doi.org/10.4213/sm602
  14. G.G. LAPTEV, Nonexistence of solutions for parabolic inequalities in unbounded cone-like domains via the test function method, J. Evol. Eq, vol.2, 42002), 459-470. https://doi.org/10.1007/PL00012600
  15. G.G. LAPTEV, Nonexistence of Solutions of Elliptic Dierential Inequalities in Conic Domains, Mat. Zametki, 71(2002), no.6, 855-866. https://doi.org/10.4213/mzm390
  16. E. MITIDIERI AND S.L. POHOZAEV, A priori estimates and blow-up of solutions to non-linear partial differential equations and inequalities, Proc. Steklov Inst. Math, 234(2001).
  17. E. MITIDIERI AND S.L. POHOZAEV, Non-existence of weak solutions for some degenerate elliptic and parabolic problems on R$^{N}$, J. Evol. Eq, 1(2001), 189-220. https://doi.org/10.1007/PL00001368
  18. A.M. PICCIRILLO, L. TOSCANO AND S. TOSCANO, Blow-up results for a class of first-order nonlinear evolution inequalities, J. Differential Equations, 212(2005), 319-350. https://doi.org/10.1016/j.jde.2004.10.026
  19. MARCO RIGOLI AND ALBERTO G. SETTI, A Liouville theorem for a class of superlinear elliptic equations on cones, Nonlinear dier. equ. appl, 9(2002), 15-36. https://doi.org/10.1007/s00030-002-8116-y
  20. NGUYEN MANH HUNG, The absence of positive solutions of second-order nonlinear elliptic equations in conical domains, Dierentsialnye Uravneniya, 34(1998), 533C539.
  21. TOMOMITSU TERAMOTOHIROYUKI USAMI, A liouville type theorem for semiliear elliptitc systems, Pacific Journal of Mathematics, Vol. 204, no.1, 2002.