By Alain Haraux
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Extra resources for Nonlinear Evolution Equations — Global Behavior of Solutions
Example text
A m o n o t o n e operator A is said to be maximal m o n o t o n e if the following p r o p e r t y holds: G(A) = G ( A ' ) A' m o n o t o n e In other words, if (n- f,$ - u) > O, [Proof. A' If [~,n] then £ H x H [~,n] I is A=A such that V[u,f] £ G(A). [$,n] £ G(A), define A' by G(A') = G(A) U [$,n]. ] This last c h a r a c t e r i z a t i o n E G(A), is sometimes useful. 4~ We state now two immediate Proposition Proof. i. For every The operator is monotone, Proposition hence 2. Then [x,y] Proof.
Glassey [8], [4]) that solutions may blow up in a finite time in the general case. The methods relying on L1apunov functions are convenient to study parabolic or hyperbolic dissipative equations. &u + ug(u 2) = 0 in H~(fl) 0 H2(fl) utt- Au + ug(u 2) + ~u t = 0 in H~(~) x L2(~) ut For example: admit respectively the Liapunov functions: [ivul2 + G(uZ)ldx and In[lut[ + IVul2 + G(u2)]dx 27 Lecture 4: Lemma 4. Methods relying on the Gronwall Bellman) (Gronwall Let lemma be a f u n c t i o n w 6 C([0,T]) >0 such that t Vt 6 [ O , T ] , w(t) <_ C + K s w(s)ds.
I FF 3u + au = ku(x,y) I x ]u(s,y)[2ds is a real constant, weight functions Concerning and the original which do not alter the nature local existence, T. Cazenave this non-standard equation was solvable space However, HI oR2,~). equation some of the problem. discovered that in the very standard in intermediate obliged to use the dissymetric involved calculations, he was space: X = L~°(IRy,L2(1Rx)). For global existence group generated by in iA played by the following H l, a decay property of the isometry was used, and a fundamental result: role was 83 Lemma 6.