Three Cylinder Inequalities and Unique Continuation Properties for Parabolic Equations (original) (raw)
Acta Mathematica Sinica, English Series, 2005
Let Γ be a portion of a C 1,α boundary of an n-dimensional domain D. Let u be a solution to a second order parabolic equation in D × (−T, T) and assume that u = 0 on Γ × (−T, T), 0 ∈ Γ. We prove that u satisfies a three cylinder inequality near Γ × (−T, T). As a consequence of the previous result we prove that if u (x, t) = O |x| k for every t ∈ (−T, T) and every k ∈ N, then u is identically equal to zero.
Remark on the strong unique continuation property for parabolic operators
2004
We consider solutions u = u(x, t), in a neighbourhood of (x, t) = (0, 0), to a parabolic differential equation with variable coefficients depending on space and time variables. We assume that the coefficients in the principal part are Lipschitz continuous and that those in the lower order terms are bounded. We prove that, if u(•, 0) vanishes of infinite order at x = 0, then u(•, 0) ≡ 0.
Unique continuation principle for systems of parabolic equations
2010
In this paper we prove a unique continuation result for a cascade system of parabolic equations, in which the solution of the first equation is (partially) used as a forcing term for the second equation. As a consequence we prove the existence of ε-insensitizing controls for some parabolic equations when the control region and the observability region do not intersect.
Proceedings of the National Academy of Sciences, 1957
This note outlines the proofs of theorems on the continuity of solutions of linear parabolic and elliptic partial differential equations. These a priori continuity
Asymptotics of solutions of second order parabolic equations near conical points and edges
Boundary Value Problems, 2014
The authors consider the first boundary value problem for a second order parabolic equation with variable coefficients in a domain with conical points or edges. In the first part of the paper, they study the Green function for this problem in the domain K × R n-m , where K is an infinite cone in R m , 2 ≤ m ≤ n. They obtain the asymptotics of the Green function near the vertex (n = m) and edge (n > m), respectively. This result is applied in the second part of the paper, which deals with the initial-boundary value problem in this domain. Here, the right-hand side f of the differential equation belongs to a weighted L p space. At the end of the paper, the initial-boundary value problem in a bounded domain with conical points or edges is studied. http://www.boundaryvalueproblems.com/content/2014/1/252
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.
On the solvability for parabolic equations with one space variable
2004
In the present work we consider the higher order linear parabolic equation in a rectangle with initial and boundary conditions. We establish new a priori estimates for the solutions to this problem in general Holder anisotropic norms, under the assumption that the coefficients and the independent term are continuous functions in the rectangle, they satisfy the general Holder condition in the rectangle of exponent a(l), with respect to the space variable only and they satisfy the general Holder condition on the boundary of exponent b(l, with respect to all variables. In this connection, however, we also obtain an estimate for the modulus of continuity with respect to the time of the higher derivatives with respect to x of the corresponding solutions. On the basis of our new a priori estimates for the solution to this problem, we establish the corresponding theorem on the solvability in general Holder anisotropic spaces. We apply our results in the linear theory to establish the local...
Capacitary estimates of solutions of semilinear parabolic equations
Calculus of Variations and Partial Differential Equations, 2013
We prove that any positive solution of ∂ t u − ∆u + u q = 0 (q > 1) in R N × (0, ∞) with initial trace (F, 0), where F is a closed subset of R N can be represented, up to two universal multiplicative constants, by a series involving the Bessel capacity C 2/q,q ′. As a consequence we prove that there exists a unique positive solution of the equation with such an initial trace. We also characterize the blow-up set of u(x, t) when t ↓ 0 , by using the "density" of F expressed in terms of the C 2/q,q ′-Bessel capacity.
Some Sharp Estimates for Parabolic Equations
Journal of Functional Analysis, 2001
Sharp estimates are given for the constants appearing in both the smoothing effect of general self-adjoint contraction semi-groups and uniform estimates for the linear heat equation. The last estimate is used to prove rather sharp global existence results for some nonlinear perturbations and suitable initial data in a quite simple and convenient way.
A regularity theorem for parabolic equations
Journal of Functional Analysis, 1971
We consider the solution in a Hilbert space H of a parabolic equation of the following type: u'(t) + A(t) u(t) = 0; 40) = %I I where A(t) is an elliptic operator depending on t. We prove, under suitable hypotheses on A(t), an abstract regularity theorem, generalising the usual result (see J. L. LIONS, "Equations DifErentielles