A critical parabolic estimate and application to nonlocal equations arising in chemotaxis

References

• Alvino , A. , Trombetti , G. and Lions , P. L. 1990 . Comparison results for elliptic and parabolic equations via Schwarz symmetrization . Ann. Inst. Henri Poincare , 7 : 37 – 65 .
   Web of Science ®Google Scholar
• Alvino , A. , Lions , P. L. and Trombetti , G. 1991 . Comparison results for elliptic and parabolic equations via symmetrization: A new approach . Diff. and Int. Eq , 4 : 25 – 50 .
   Google Scholar
• Bandle , C. 1976 . On Symmetrization in parabolic equations . J. 'Anal. Math , 30 : 98 – 112 .
   Web of Science ®Google Scholar
• Biler , P. and Nadzieja , T. 1994 . Existence and nonexistence of solutions for a model of gravitational interaction of particles I . Coll Math , 66 : 319 – 334 .
   Google Scholar
• Biler , P. , Hilhorst , D. and Nadzieja , T. 1994 . Existence and nonexistence of solutions for a model of gravitational interaction of particles II . Coll Math , 67 : 297 – 308 .
   Google Scholar
• Brezis , H. and Merle , F. 1991 . Uniform estimates and blow up behavior for solutions of in two dimensions . Comm. P.D.E , 16 : 1223 – 1253 .
   Web of Science ®Google Scholar
• Chen , W. and Li , C. 1993 . Apriori estimates for solutions to nonlinear elliptic equations . Arch. Rat. Mech. Anal , 122 : 145 – 157 .
   Google Scholar
• Childress , S. 1984 . Chemotactic collapse in two dimensions . Lecture Notes in Biomath , 55 : 61 – 66 .
   Google Scholar
• Childress , S. and Percus , J. K. 1981 . Nonlinear aspects of chemotaxis . Math. Bioscience , 56 : 217 – 237 .
   Web of Science ®Google Scholar
• Davis , E. B. 1990 . "Heat Kernels and Spectral Theory" ” . Cambridge Univ. Press .
   Google Scholar
• Gidas , B. , Ni , W. M. and Nirenberg , L. 1979 . Symmetry and related properties via the maximum principle . Comm. Math. Phys , 68 : 209 – 243 .
   Web of Science ®Google Scholar
• Herrero , M. A. and Velázquez , J.J.L. in press . Singularity patterns in a chemotaxis model .
   Google Scholar
• Jager , W. and Luckhaus , S. 1992 . On explosion of solutions to a system of partial differential equations modeling chemotaxis . Trans. Amer. Math. Soc , 392 : 819 – 824 .
   Google Scholar
• Kazdan , J. L. and Warner , F. W. 1967 . Curvature functions for compact 2 manifolds . Ann. Math , 99 : 14 – 47 .
   Google Scholar
• Keller , E. F. and Segel , L. A. 1971 . Traveling bands of Chemotactic Bacteria: A theoretical analysis . J. Theor. Biol , 30 : 235 – 248 .
   PubMed Web of Science ®Google Scholar
• Li , Y. Y. and Shafir , I. 1994 . Blow up analysis for solutions of in dimension two . Ind. Univ. Math. J , 43 : 1255 – 1270 .
   Web of Science ®Google Scholar
• Ladyzenskaja , O. A. , Solonnikov , V. A. and Ural'ceve , N. N. 1968 . Linear and Quasilinear Equation of Parabolic Type . Trans of Math. Monographs , 23
   Google Scholar
• Nagai , T. 1995 . Blow up of radially symmetric solutions to a chemotaxis system . Advances in Math. Science and Appl , 5 : 581 – 601 .
   Google Scholar
• Nanjundiah , V. 1973 . Chemotaxis, signal relaying and aggregation morphology . J. Theor. Biol , 42 : 63 – 105 .
   PubMed Web of Science ®Google Scholar
• Pinchover , Y. 1991 . Large time behavior of the heat kernel and the behavior of the Green function near criticality for nonsymmetric elliptic operators . J. Func. Anal , 104 : 54 – 70 .
   Google Scholar
• Wolansky , G. 1992 . On steady distributions of self attracting clusters under friction and fluctuations . Arch. Rational Mech. Anal , 119 : 355 – 391 .
   Web of Science ®Google Scholar
• Wolansky , G. 1992 . On the evolution of self interacting clusters and applications to semi linear equations with exponential nonlinearity . J. d'Analyse Math , 59 : 251 – 272 .
   Google Scholar