Download PDF by Jerrold Bebernes: Mathematical Problems from Combustion Theory

By Jerrold Bebernes

ISBN-10: 0387971041

ISBN-13: 9780387971049

ISBN-10: 146124546X

ISBN-13: 9781461245469

ISBN-10: 146128872X

ISBN-13: 9781461288725

This monograph developed during the last 5 years. It had its beginning as a collection of lecture notes ready for the 9th summer season university of Mathematical Physics held at Ravello, Italy, in 1984 and used to be additional sophisticated in seminars and lectures given basically on the collage of Colorado. the cloth awarded is the fabricated from a unmarried mathematical query raised through Dave Kassoy over ten years in the past. this query and its partial solution ended in a winning, fascinating, virtually detailed interdisciplinary col­ laborative clinical attempt. The mathematical types defined are frequently instances deceptively easy in visual appeal. yet they show a mathematical richness and wonder that belies that simplicity and affirms their actual importance. The mathe­ matical instruments required to solve many of the difficulties raised are various, and no systematic try is made to offer the mandatory mathematical historical past. The unifying subject of the monograph is the set of types themselves. This monograph might by no means have come to fruition with out the enthu­ siasm and force of Dave Eberly-a former pupil, now collaborator and coauthor-and with out numerous major breakthroughs in our comprehend­ ing of the phenomena of blowup or thermal runaway which yes types mentioned own. A collaborator and previous scholar who has made major contribu­ tions all through is Alberto Bressan. there are various different collaborators­ William Troy, Watson Fulks, Andrew Lacey, Klaus Schmitt-and former students-Paul Talaga and Richard Ely-who needs to be said and thanked.

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We have u(x) ~ 0 (where u(O) = 0) and u( -x) = u(x), but u(x) is not radially decreasing. Note that the condition f(u) ~ 0 for all u implies that any nontrivial solution is positive (by the maximum principle). Although the result is stated for f E C 1, this hypothesis can be weakened. The result also holds for any function f = h + h where h E C 1 and h is monotone increasing. In particular, the result holds if f is locally Lipschitz continuous. The proof utilizes maximum principles and the method of moving parallel planes.

20) points strictly outward from T on both L+ and L_. In particular, (x 0 , x0 j(n- 2)) E E+ and (x 0 , 0) E E_, so these sets are nonempty. By continuous dependence, the sets E+ and E_ are open sets (relative to L). Moreover, E+ n E_ = 0 and E+ n E_ = 0. Since Lis a connected set, L -:f. E+ U E_. That is, there must be at least one point (xo, Yo) whose flow t(Yo) meets (0, 0) at t = oo. Uniqueness. 20) has the property x(t) < 0, so by the Inverse Function Theorem, one can think oft = t(x) and y = y(x).

29) has a solution u(x, t) on 0 x [0, T] for any other domain 0 of the same volume, and 2. max{ u(x, t) : x E 0} ~ max{O(x, t) : x E BR} for all t E [0, T]. 8), we have the following result. 12 Consider IBVP {9. 8) with f(u) > 0, f'(u) ~ 0, and f"(u) ~ 0 for u ~ 0, and f 00 [f(u)]- 1du < oo. 1 sup{ u/ f(u) : u ~ 0}, then the unique solution O(x, t) of {9. 3.

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