The Cauchy Problem for Strictly Hyperbolic Operators with Non-Absolutely Continuous Coefficients (original) (raw)
Blow-up results for some nonlinear hyperbolic problems
We also study the nonexistence for some more general inequalities of second order but without any assumption about the type of operator. The method relies on a suitable choice of test functions, resealing techniques and a dimensional analysis.
Qualitative properties for some nonlinear higher order degenerate parabolic equations
1988
We prove that if 1 < q < p the "energy solutions" of 0 [q_• u) +(1)• Z D• ([D•ulp-•sgnD•u) f 07 (lu sgn-have finite speed of propagation. For p-2 the second term is (-A)"•u. We also study upper and lower bounds of the interface (or free boundary), decay rates as t-• c•, extinction in finite tinhe, nonexistence of non-negative solutions and some generalizations to other equations. 319 32O F. BERNIS We prove that if 1 < q < p the energy solutions (Definition 2.1) of (1.1)-(1.4) have finite speed of propagation (Theorem 2.1 and corollaries). This theorem also gives estimates on the growth of the interface (or free boundary) defined by support u(.,t). (In fact, we study the "outer" interface, since support u(., t) may be nonconnected. See (2.4) for a precise concept.) In Section 3 we give the formulae of integration by parts used to obtain energy estimates. Section 4 consists in the proof of Theorem 2.1. We use a weighted energy method, adapted from [12], the weights being powers of the distance to a v•riable half-space. The main technical tools are some weighted Gagliardo-Nirenberg inequalities (see Appendix I). We do not use regularity results, nor comparison principles (which are not valid for m > 2), nor explicit solutions (which are not known for m > 2). No hypothesis on the sign of u0 is made. This gives some generality to the method (see Section 11). We take f = 0 in Sections 5 to 8 and 10. Section 5 gives decay rates as t-• cx• for the Lq norm of u(-, t). For m = I the power groxvth of the interface (of exponent /•0, see (2.8)) obtained from Theorem 2.1 is greater (as t-• cx•) than the known Barenblatt rate (of exponent /•, see (1.7)). In Section 6 we establish that (for any m) a "Barenblatt-like" rate/• (see (6.3)) as t-• cx• is equivalent to the knowledge of an L•(0,•; Lq-•(•)) estimate of u (which corresponds to an L•(0, ec;L•(•)) estimate in the usual setting of the porous media equation; see below). In Section 7 (f• = R n, support u0 bounded, u0 • 0) we prove the nonexistence of global nonnegative solutions if m > 3 (m > 2 for p = 2). This is in sharp contrast with the second order case. The bounds of the former sections are upper bounds. The results on "lower bounds" (as well as Section 7) rely on the invariance (in time) of certain x-moments of u •-• sgn u for f• = R n or for support u bounded away from the boundary (see Theorems 7.1 and 7.2 and Remark 7.1). In Section 8 xve prove that support u(-, t) expands unboundedly as t-• cx• if at least one of the invariant moments is different from zero (Theorem 8.1, • = R•). The Barenblatt-like rate/• is both an upper and a lower bound under the conditions stated in Theorem 8.2. The rate/•0 of Theorem 2.1 is, roughly speaking, optimal for solutions of finite energy if n <mp (Theorem 9.1, • = R •, f •: 0 allowed). Section 10, which is closely related to Section 5, deals with the property of extinction in finite time. It is the only section with q > p. Some QUALITATIVE PROPERTIES FOR SOME PARABOLIC EQUATIONS 321 variations of equation (1.1) are considered in Section 11. Theorem 2.1 extends to some nonhomogeneous Dirichlet data, as explained in Remark 4.3. On the contrary, in Section 5, 6 and 10 condition (1.2) is essential. The porous media equation. The case ra = 1, p = 2, f = 0, of (1.1) is related to the porous media equation (1.5) •v ot-(Iv v): o by the change v = u q-1 sgn u, q-1 = 1/•I. We note that g'/ > 1 is equivalent to 1 < q < 2 and v(.,t) 6 L • is equivalent to u(.,t) 6 L q-1. There is a very extensive literature on (1.5)' see the surveys of Peletier [36] and Aronson [4] and other references beloxv. Related work. To our knowledge, the above results are ne;v for order >_ 4, i.e., for ra _> 2. For m = I these results are sharper in many cases. The property of finite speed of propagation for (1.1) (1.4), ra = 1, ;vas obtained in the following works: Oleinik, Kalashnikov & Yui-Lin [34] for p = 2, n-1; Diaz [lS] for io = 2, Vn; Kalashnikov [28] for io y• 2, n = 1; Diaz & Iterrero [20] for q = 2, Vn; Dfaz &; V•ron [21], [22] for p 5• 2, q 5• 2, Vn. We refer to Diaz [19] for a survey on this and other extinction properties. Most of second order literature on finite speed of propagation uses the comparison method introduced by Brezis & Friedman [16]. Antoncev [3] and Diaz &; V•ron [21], [22] already use an energy method and imbedding-interpolation inequalities. (The use of energy methods to study the t-behaviour is more widespread: see Remarks 5.1, 10.1 and 10.2.) References on other qualitative properties will be given in the corresponding sections. Several concepts of solution are used in the quoted literature. Barenblatt explicit solutions for the second order case. For any a > 0, the function 1(•in__l Xi P') (p-1)/(p-q U(x, t) = t-•-• a-b t/•,p, + pt where (s)+ = max{s,0}, p' = p/(p-1), oq = n/•l/(q-1), b=/•-•(p-q)/p and (1.7) /•1-(P+ n(p-q)/(q-1))-1 Banach, Ann. Pac. Sci. Toulouse I (1979), 171-200. 40. L. A. Carfarelit and A. Friedman• Regularity of the free boundary of a gas flow in an n-dimensional porous medium, Indiana Univ. Math. J. 29 (1980), 361-391.
Annales de l'Institut Henri Poincare (C) …, 2011
We consider blow-up solutions for semilinear heat equations with Sobolev subcritical power nonlinearity. Given a blow-up pointâ, we have from earlier literature, the asymptotic behavior in similarity variables. Our aim is to discuss the stability of that behavior, with respect to perturbations in the blow-up point and in initial data. Introducing the notion of "profile order", we show that it is upper semicontinuous, and continuous only at points where it is a local minimum. Résumé Nous considérons des solutions explosives de l'équation semilinéaire de la chaleur avec une nonlinéarité sous-critique au sens de Sobolev. Etant donné un point d'explosionâ, grâceà des travaux antérieurs, on connaît le comportement asymptotique des solutions en variables auto-similaires. Notre objectif est de discuter la stabilité de ce comportement, par rapportà des perturbations du point d'explosion et de la donnée initiale. Introduisant la notion de "l'ordre du profil", nous montrons qu'il est semicontinu supérieurement, et continu uniquement aux points où il est un minimum local.
A Sufficient Condition for Slow Decay of a Solution to a Semilinear Parabolic Equation
Analysis and Applications, 2012
On considere l'équation ψ t − ∆ψ + c|ψ| p−1 ψ = 0 avec les conditions aux limites de Neumann dans un ouvert connexe borné de R n avec p > 1, c > 0 . On montre que si la donnée initiale est petite en norme L ∞ et si sa moyenne dépasse en valeur absolue un certain multiple de la puissance p de sa norme L ∞ , alors ψ(t, ·) décroit comme t − 1 (p−1) .