On the support of solutions to the g-KdV equation (original) (raw)

On the regularity of solutions to the 𝑘-generalized Korteweg-de Vries equation

Proceedings of the American Mathematical Society, 2016

This work is concerned with special regularity properties of solutions to the k k -generalized Korteweg-de Vries equation. In [Comm. Partial Differential Equations 40 (2015), 1336–1364] it was established that if the initial datum is u 0 ∈ H l ( ( b , ∞ ) ) u_0\in H^l((b,\infty )) for some l ∈ Z + l\in \mathbb {Z}^+ and b ∈ R b\in \mathbb {R} , then the corresponding solution u ( ⋅ , t ) u(\cdot ,t) belongs to H l ( ( β , ∞ ) ) H^l((\beta ,\infty )) for any β ∈ R \beta \in \mathbb {R} and any t ∈ ( 0 , T ) t\in (0,T) . Our goal here is to extend this result to the case where l > 3 / 4 \,l>3/4 .

On the support of solutions to the generalized KdV equation

Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2002

It is shown that if u is a solution of the initial value problem for the generalized Korteweg-de Vries equation such that there exists b ∈ R with supp u(•, t j) ⊆ (b, ∞) (or (−∞, b)), for j = 1, 2 (t 1 = t 2), then u ≡ 0.  2002 Éditions scientifiques et médicales Elsevier SAS

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.