ON THE PERTURBATION PROPAGATION IN THE INITIAL-BOUNDARY VALUE PROBLEM FOR QUASILINEAR FIRST ORDER EQUATIONS in the domain with the conditions

ON THE PERTURBATION PROPAGATION IN THE INITIAL-BOUNDARY VALUE PROBLEM FOR QUASILINEAR FIRST ORDER EQUATIONS

In Section 2 the definition of generalized solution and comparison theorem are given.
In Section 3 we deal with the case when there exists T < +oo such that ui (T -0) = +oo.According to the terminology of [1], [5] it corresponds to the so-called "processes with aggravation" .Definition 1.1.One says that localization in the problem (1 .1),(1 .2) occurs if there exists X > 0 such that u(t, x) -0 for x >_ X, 0 <_ t <_ T. One says that localization does not occur if for every sufficiently large x,k > 0 there exists t* > 0 such that u(t* , x,,) :,~0.
In the paper [1] autonomous equations with power nonlinearities and zero lower order term were studied .There necessary and sufficient cónditions for the occurence of localization and for inner boundedness of solutions were obtained.In Section 3 we shall study such questions for arbitrary nonlinearities and in the nonautonomous case.
Section 4 is devoted to localization in the case when ui (t) is defined for every t E [0, +oo) and may tend to zero as t => +oo.
Some supplementary results on the localization are given in Section 5 for the equation There are certain peculiarities of the front behavior in this case.
For the proof of this theorem similar methods to those of papers [2], [10] are used.The uniqueness of the g.s. for problem (1.1), (1 .2) follows from Theorem 2 .1 .
Proof. .Suppose the line x = y(t) is defined by the equations

Proof. . Let us consider the function
The equation G(t, v) = 0 with respect to v has two roots : v = 0, v = vl(1/(T -t)) .When x varies the solution of (3.2) may stop to exist if G (t, v) = 0. Consequently the set of (t, x) where the solution of (3.2) does not exist can be described by the system Now, let us consider the function y(t) defined in the following way y = A (t, y, wo(t, y))/wo(t, y), y(0) = 0 .Then y :~A v(t, y, wo) ::~a o(T -t)a(wo) < g(T -t)a(v) .
Proof.Let us consider the function wo(t, x) introduced in the proof of Theorem 3 .2 .
Suppose y(t) is defined by the equation y = A(t, y, wo(t, y))/wo(t, y) with the initial datum y(0) = 0.By analogy with the proof of Theorem 3 .2one states that the curve x = y(t) is contained in the domain of existente of the solution to equation (3.2) .Let us regard the same comparison function \2(t,x) as in the proof of Theorem 3 .2 .As G, > 0 for x < z(t) one has Lwo < 0 for x < y(t) and U (t, x) > >12 (t, x) in Q.
Now the equation G(t, v) = 0 has two roots and the root of the equation Gv = 0 lies between them by virtue of Rolle's Theorem .Consequently the solution v > 0 of the equation (3.2) with fixed x, t always exceeds the solution of the equation Gv = 0 with the same fixed t.Hence Using conditions 7) one estimates : [s-1bo(1 /s)cp(s) + sw (s)] ?VI(S)04 (8) ; Since p(s) and w1(s) do not increase v(s) does not increase too.

Y . G. RYKOV
Then u(t, x) is unbounded as t ==> T -0 for every fixed x.

The case T = +oo
For the space of this paragraph we asume T = +oo, 0 <_ uI (t) <_ M for t E 1[8+ .
It is easy to see that LA4 < 0 at the points where \4(t, x) is smooth while at the line of discontinuity x = y(t) relations (2.1), (2 .2) are valid.With the aid of Theorem 2 .1 it follows that u(t, x) > A4(t, x) in 1[8+ x R+.
T -0.This ends the proof.y(t*) = Remark 3.4.In the case of equation (3.1) Theorem 3.4 states the absence of localization for p < -1.