By P. I. Naumkin and I. A. Shishmarev

This publication is the 1st to pay attention to the idea of nonlinear nonlocal equations. The authors clear up a few difficulties about the asymptotic habit of options of nonlinear evolution equations, the blow-up of options, and the worldwide in time lifestyles of suggestions. furthermore, a brand new class of nonlinear nonlocal equations is brought. a wide classification of those equations is taken care of via a unmarried process, the most good points of that are apriori estimates in several essential norms and use of the Fourier remodel. This ebook will curiosity experts in partial differential equations, in addition to physicists and engineers.

Readership: experts in partial differential equations.

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For n > 0 we denote jn = 2 f (p)2p2fldp, A _ (n+ 1)°(2"-5), o' = 1/2. TxeoxEivt 3. 31) Ana-2n-1, n>0 with a sufficiently large constant a, a> c(T, b, a). 1) belonging to C°°([0, T];H°°(Ri)). PROOF. 1) with the operators 38 2. 34) Ana-211 where J,,(t) t - 2n 0(211-5), ul(p, n = 34 ... 34) hold for t = 0. 22) from the previous subsection. N' > 3. N' Assuming the contrary, let Tl > 0 be the first moment of time when one of these inequalities is violated. 33). IZ <42a3. 12 < 5 JZ + 2bA2a-4egr (1 + 4a-«Z- "2 ) a < A2a-4(qe` + 8ba-ae`1TZ-1'2) .

We c note that the time T of existence of a classical solution is determined by the fact that the function pi (t) is bounded. 9), the functions v(t) are also bounded. 1) with a regular operator K can be extended in time as long as the first derivative with respect to x of the solution u(x, t) remains bounded. Thus, destruction of the classical solution can occur only as a result of breaking. The theorem is proved. 2. Dissipative operator. 1) in the case of a dissipative operator K. 1) does not belong to L1 (R1) and is, for instance, a distribution).

1) prevails over the integral term. THEoxEM 3. Let the kernel k(x) satisfy the following conditions: 1) k(x) E C2(R1\0), f>a I< oo, a> 0, j,p = 1,2; Ik'(x)I coIxI-'-', x 0, a = 3 - Y, 0 0; 2) Ik(x)l 3) there exists an interval (-cel, ce2), ai, a2 >0, such that k' (x) < 0 for x E (-ai,a2), Then, if Im(0)I is sufficiently large, the wave u(x, t) breaks at the moment of time 1 Im(0)I(1 - y/8) We introduce the notation f 100 2() k()u(xi(t) - / Z (t) = 1 + m(0) I t \ f dt'\ A(t') I with m(t) and xl (t) as in Theorem 2.

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