Jerry, Goldstein

English

Embry-Riddle Aeronautical University

Instantaneous blowup, old and new resultsJerry Goldstein, University of MemphisConsider@u@t= u+cjxj2u+ f(x; t);x 2 RN ; t  0;u(x; 0) = u0(x):where f  0; u0  0 and (f; u0) 6= 0: Assume u0 and f are such that aunique positive solution exists for all t  0 if c = 0: Then for c  C(N) = N 222;many global (in time) solutions exist, but for c > C(N); no positivesolution in the sense of distributions exists. In fact, if V (x) = c= jxj2 is replacedby Vn(x) = minfV (x); ng and if f(x; t) is replaced by fn(x; t) = minff(x; t); ng;and if un is the corresponding unique positive solution, then un(x; t) ! 1 asn ! 1 for all x 2 RN ; t > 0: This is instantaneous blowup (IBU). This 1984theorem of Pierre Baras and Jerry Goldstein led to much additional research onsingular linear and nonlinear parabolic PDE. We will discuss new proofs andnew theorems, including IBU when RN is replaced by the Heisenberg groupHN ; nonexistence of positive solutions (even locally in time) for certain non-linear parabolic problems, and related topics, involving the Ornstein-Uhlenbeckequation and others. . This is joint work with many coauthors.1

Instantaneous Blowup, Old and New Results

https://commons.erau.edu/cgi/viewcontent.cgi?article=1112&context=searcde

Instantaneous Blowup, Old and New Results

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