Download e-book for iPad: Nonlinear Differential Equation Models by Björn Gustafsson, Makoto Sakai (auth.), Prof. Dr. Ansgar

By Björn Gustafsson, Makoto Sakai (auth.), Prof. Dr. Ansgar Jüngel, Prof. Dr. Raul Manasevich, Prof. Dr. Peter A. Markowich, Prof. Dr. Henrik Shahgholian (eds.)

ISBN-10: 3709106095

ISBN-13: 9783709106099

ISBN-10: 370917208X

ISBN-13: 9783709172087

The papers during this booklet originate from lectures which have been held on the "Vienna Workshop on Nonlinear types and research" – may possibly 20–24, 2002. They characterize a cross-section of the examine box utilized Nonlinear research with emphasis on unfastened limitations, absolutely nonlinear partial differential equations, variational equipment, quasilinear partial differential equations and nonlinear kinetic models.

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Example text

Proposition 13. There exists a solution w of (5) such that v(-, t) ---+ we) as t ---+ 00 uniformly on [0,00). Proof Let tn be a sequence such that tn ---+ 00. From Lemma 10, Lemma II and Ascoli-Arze1a's theorem there exists a subsequence, that we still call tn, and a solution wof(5) such that v(·, tn ) ---+ w(·) uniformly, as n ---+ 00. We recall that w == orw = w. Now we have two possibilities, either every sequence has a subsequence that converges to wand in this case we have v(·, t) converge to w as t ---+ 00.

Let t> O. Then for any 8> 0, sufficiently small, there exists t E [t, t + 8] and a constant C, that depends only on m and uo, such that v(O,""i) ~ C8-~. On the Blow-Up Set For u, = (um)xx' m> I 51 Proof Integrating in (4) we have +OO VI(X, t) dx = J+oo (vmtx(x, t) dx - - -11 J+oo v(x, t) dx 0 m0 Jo 1 J+oo = vm(O, t) - --1 v(x, t) dx. m- 0 Integrating again we get I+6 J+oo vI(x,s)dxds= JI+6 vm(O,s)ds--_1 JI+6 J+oo v(x,s)dxds. m 1 0 J 0 1 1 1 Hence I+6 vm(O, s) ds = -1- JI+6 J+oo v(x, s) dxds + J+oo [v(x, t + 8) m-1 0 0 J 1 v(x, t)] dx.

Lemma 3. Let Ii be a classical sub-solution of (2) and assume that u(O, 0) and Ii(X,O) < u(x, 0) > 0, for all x E [0, 00). Then Ii ~ u in DT· We will also use the following integral comparison lemma. Lemma 4. Let u be a super-solution and Ii a sub-solution of (2) such that J~ Ii(Y, 0) dy ~ J~ u(y, 0) dy and J~ Ii(Y, 0) dy + L-(Iim)AO, ~ J~ u(y, 0) dy + L for all x E [0,00), s) ds -(um)AO, s) ds for all t E [0, T). Then J~ Ii(Y, t) dy ~ J~ u(y, t) dy in D T · 4. Blow-Up in Finite Time and Localization We will prove first that every non-trivial solution blows up in finite time.

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Nonlinear Differential Equation Models by Björn Gustafsson, Makoto Sakai (auth.), Prof. Dr. Ansgar Jüngel, Prof. Dr. Raul Manasevich, Prof. Dr. Peter A. Markowich, Prof. Dr. Henrik Shahgholian (eds.)


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