scholarly journals A priori bounds and complete blow-up of positive solutions of indefinite superlinear parabolic problems

2005 ◽  
Vol 304 (2) ◽  
pp. 614-631 ◽  
Author(s):  
Pavol Quittner ◽  
Frédérique Simondon
Keyword(s):  
Positive Solutions ◽  
Blow Up ◽  
A Priori ◽  
Parabolic Problems ◽  
A Priori Bounds

Positive Solutions for Systems of Quasilinear Equations with Non-homogeneous Operators and Weights

2020 ◽  
Vol 20 (2) ◽  
pp. 293-310
Author(s):  
Marta García-Huidobro ◽  
Raúl Manasevich ◽  
Satoshi Tanaka
Keyword(s):  
Positive Solutions ◽  
Differential Operators ◽  
Blow Up ◽  
A Priori ◽  
Degree Theory ◽  
Radial Solutions ◽  
A Priori Bounds ◽  
Existence Of A Solution ◽  
Weakly Coupled System ◽  
Positive Radial Solutions

AbstractIn this paper we deal with positive radially symmetric solutions for a boundary value problem containing a strongly nonlinear operator. The proof of existence of positive solutions that we give uses the blow-up method as a main ingredient for the search of a-priori bounds of solutions. The blow-up argument is one by contradiction and uses a sort of scaling, reminiscent to the one used in the theory of minimal surfaces, see [B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations 6 1981, 883–901], and therefore the homogeneity of the operators, Laplacian or p-Laplacian, and second members powers or power like functions play a fundamental role in the method. Thus, when the differential operators are no longer homogeneous, and similarly for the second members, applying the blow-up method to obtain a-priori bounds of solutions seems an almost impossible task. In spite of this fact, in [M. García-Huidobro, I. Guerra and R. Manásevich, Existence of positive radial solutions for a weakly coupled system via blow up, Abstr. Appl. Anal. 3 1998, 1–2, 105–131], we were able to overcome this difficulty and obtain a-priori bounds for a certain (simpler) type of problems. We show in this paper that the asymptotically homogeneous functions provide, in the same sense, a nonlinear rescaling, that allows us to generalize the blow-up method to our present situation. After the a-priori bounds are obtained, the existence of a solution follows from Leray–Schauder topological degree theory.

scholarly journals A priori bounds, nodal equilibria and connecting orbits in indefinite superlinear parabolic problems

2008 ◽  
Vol 360 (07) ◽  
pp. 3493-3540 ◽  
Author(s):  
Nils Ackermann ◽  
Thomas Bartsch ◽  
Petr Kaplický ◽  
Pavol Quittner
Keyword(s):  
A Priori ◽  
Parabolic Problems ◽  
A Priori Bounds ◽  
Connecting Orbits

scholarly journals Life Span of Positive Solutions for the Cauchy Problem for the Parabolic Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-16
Author(s):  
Yusuke Yamauchi
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equation ◽  
Life Span ◽  
Recent Result ◽  
Positive Solutions ◽  
Parabolic Equations ◽  
Blow Up ◽  
Parabolic Problems ◽  
Survey Paper ◽  
The Cauchy Problem

Since 1960's, the blow-up phenomena for the Fujita type parabolic equation have been investigated by many researchers. In this survey paper, we discuss various results on the life span of positive solutions for several superlinear parabolic problems. In the last section, we introduce a recent result by the author.

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