ON NONEXISTENCE OF SOLUTIONS TO SOME NONLINEAR PARABOLIC INEQUALITIES

. We obtain suﬃcient conditions for nonexistence of positive solutions to some nonlinear parabolic inequalities with coeﬃcients possessing singu- larities on unbounded sets.


1.
Introduction. The problem of sufficient conditions for nonexistence of solutions to nonlinear parabolic differential equations and inequalities with singular coefficients was studied by many authors.
For the Laplacian and heat operator with a point singularity inside the domain, pioneering results in this direction were obtained by H. Brezis and X. Cabre [1] by means of comparison principles.
For higher order operators that do not satisfy the comparison principle, S. Pohozaev [6] suggested the nonlinear capacity method. Later it was developed in joint works with E. Mitidieri and other authors (see, in particular, monograph [5] and references therein). This method allowed one to obtain a number of new sharp sufficient conditions of non-solvability of differential inequalities in various functional classes. The method is based on deriving asymptotically optimal a priori estimates of the solutions by means of algebraic analysis of the integral form of the inequality under consideration with a special choice of test functions. Some applications of this method to nonlinear parabolic inequalities with singular coefficients at a single point or on the boundary can be found in [2] and [7,8].
In the present paper, a modification of the nonlinear capacity method is used in order to obtain sufficient conditions of non-solvability for nonlinear parabolic inequalities both of second and of higher order, including those with nonlinear principal terms and, most importantly, with coefficients having singularities on some unbounded sets inside the domain of definition. This distinguishes the problem setting suggested here from the previous works in this field, where singularities appeared on certain bounded sets instead.
For the proof of nonexistence results by the nonlinear capacity method, test functions with different geometrical structure of the support are constructed, which takes into account the specific nature of problems under consideration. Our first results in this direction were published in [3,4].
The rest of the paper consists of four sections. In §2, we formulate geometrical assumptions on the set where the coefficients of the problems have singularities, and construct appropriate families of test functions. In §3, we establish nonexistence results for higher order semilinear parabolic inequalities, and in §4, for second order quasilinear ones.
Remark on notation. From here on, letter c denotes different positive constants, which may depend on the parameters of the problems under consideration. Letters c 0 , c 1 , . . . denote absolute positive constants. For q > 1, we denote by q the conjugate exponent defined by 1 q + 1 q = 1 (that is, q = q q−1 ).

2.
Assumptions on the singular set and test functions. We consider coefficients which have singularities on a closed possibly unbounded set S ⊂ R n , which has geometrical structure characterized by the following assumptions. Let ε > 0. Denote Assume that: (H 1 ) There exist constants c 0 > 0 and θ > 0 such that for sufficiently large R > 0 one has (H 2 ) For some k ∈ N, there exist a family of functions ξ R ∈ C 2k 0 (R n \ S; [0, 1]) such that and a constant c > 0 such that for all multi-indices α with 0 ≤ |α| ≤ k.

Example 1.
As the set S one can consider a hyperplane S = Π n = {x = (x 1 , . . . , x n ) ∈ R n : x n = 0}. Then for all sufficiently large R > 0 one has where m n−1 is the measure of a unit ball in R n−1 . Thus Π n satisfies assumption (H 1 ) with c 0 = 2m n−1 and θ = 2. Similar estimates hold for any hyperplane in R n . For l-dimensional planes (1 ≤ l ≤ n − 1), assumption (H 1 ) holds with θ = n − l.
In some of our results, we will require that the following assumptions are valid: (H * 1 ) There exist positive constants c 1 , c 2 , θ such that for sufficiently large R > 0 one has and (3) holds for all multi-indices α with 0 ≤ |α| ≤ k.
To verify (H * 2 ) for S = Π n , one can take ξ 1

846
OLGA SALIEVA We will also use functions ψ R ∈ C k 0 (R n ; [0, 1]) such that and a constant c > 0 such that 3. Semilinear parabolic inequalities of higher order. A nonexistence result takes place for the semilinear parabolic inequality with the initial condition Define weak solutions of the Cauchy problem (9)-(10) in the following way.
for some t * > 0, for all t 0 , t 1 such that 0 ≤ t 0 < t 1 ≤ t * , and for any nonnegative , provided that all integrals exist and lim Remark 1. If u and A α are sufficiently regular, inequality (11) can be derived from (9)-(10) by integration by parts.
Sufficient conditions for nonexistence of solutions to the Cauchy problem (9)-(10) in the given sense can be formulated as follows.
Theorem 3.2. Let the set S satisfy conditions (H 1 ) and (H 2 ). Suppose that the initial function u 0 ∈ C(R n \ S) is nonnegative, a > 0, and the coefficients A α : R n × R → R are Carathéodory functions such that for all α : |α| ≤ k there exist functions that satisfy the estimate and Then problem (9) − (10) has no global nonnegative solutions u in (R n \ S) × R + that are distinct from the identical zero.
Proof. For the Cauchy problem (9)-(10) we introduce test functions ) κ depend only on spatial variables and T τ (t) on time. Here ξ R are defined as in (H2), ψ R satisfy (7)-(8), κ > kq q−1 , and T τ ∈ C 1 ([0, τ ]; [0, 1]) with τ > 0 are such that and besides with some constant c > 0. Multiplying both parts of (9) by ϕ R (x)T τ (t) and integrating in parts, we get Applying the parametric Young inequality to both terms on the right-hand side, we arrive at a 2 Due to the choice of ϕ R (x) and T τ (t), we can restrict integration to smaller domains on both sides of the inequality: Note that the second term on the left-hand side is nonnegative and ϕ R (x) ≡ 1 in the whole integration domain, and the first integral on the right-hand side is estimated by condition (12). Representing ϕ R (x) ≡ ξ R (x)ψ R (x) and using (3), (8) and (14) with r = q , we get that which due to (12) for sufficiently large R implies One can easily see that the right-hand side of (15) attains its minimum at Substituting (16) into (15) and taking R → ∞, under assumption (13) we arrive at a contradiction, which proves the claim.
Under additional assumptions on the behavior of the initial function we can obtain sufficient conditions for nonexistence not only for global solutions of problem (9) − (10) but also for local ones. Namely, there holds Theorem 3.3. Let the set S satisfy conditions (H * 1 ) and (H * 2 ), k ∈ N, q > 1, and the initial function u 0 ∈ C(R n \ S) satisfies the inequality with some constants c 0 > 0 and µ ∈ R, where Then the Cauchy problem (9)−(10) has no positive solutions u in (R n \S)×[0, T ] for any arbitrarily small T > 0.

Proof. Repeating the previous arguments with functions
, similarly to (15) we obtain the estimate One can easily see that the right-hand side of (19) attains its minimum at Substituting (20) into (19), estimating the left-hand side of the obtained inequality from below with the help of (17) and passing to the limit as R → ∞, under assumption (18) we reach a contradiction, which proves the claim.
4. Quasilinear parabolic inequalities. Let u 0 ∈ C(R n \ S), c > 0. Then one can formulate the Cauchy problem We define weak solutions of the Cauchy problem (21) in the following way.
for some t * > 0, for all t 0 , t 1 such that 0 ≤ t 0 < t 1 ≤ t * , and for any nonnegative function φ ∈ C 1 ((R n \ S) × [t 0 , t 1 ]) such that for all t ∈ [t 0 , t 1 ] one has φ(·, t) ∈ C 1 0 (R n \ S), provided that all integrals exist and lim with some constants c 1 , c 2 > 0, the initial function u 0 ∈ L 1 loc (R n \ S) is nonnegative, Then the Cauchy problem (21) has no global nonnegative solutions u in (R n \S)×R + that are distinct from the identical zero.
Proof. For simplicity consider A(x, u, Du) = |Du| p−2 . Suppose that a solution u of problem (21) does exist and consider the weak formulation (22) with test functions , ξ R are as in (H2), ψ R satisfy assumptions (7)- (8), and for T τ one has (14) with r = q + ν q − 1 , where ν < 0 and |ν| is sufficiently small. In this case inequality (22) takes the form Integrating the last term in this inequality by parts twice, we obtain Therefore (26) can be rewritten as Further we apply the Young inequality with appropriate parameters to the second and third terms on the right-hand side: Similarly, the second term on the right-hand side of (28) can be estimated as Since ϕ R = (ξ R · ψ R ) κ , due to (3) and (8) for κ > pq q−p+1 we have and similarly Combining inequalities (27)-(32), we obtain Here we take into account that ν < 0. Since u 0 is nonnegative, we can omit the second term on the left-hand side. This and (14) imply where c > 0 and By virtue of (24), the term R −θ−lν can be omitted for sufficiently large R and small |ν|. Then one can easily see that the right-hand side of (34) attains its minimum at where the exponent is positive by assumptions (23) and q > p − 1. Substituting (36) into (34), we reach a contradiction as R → ∞ due to assumption (25) if |ν| is sufficiently small. This completes the proof.
Under additional assumptions on the behavior of the initial function we can obtain sufficient conditions for nonexistence not only for global solutions of problem (21) but also for local ones. Namely, there holds Theorem 4.3. Let S satisfy (H * 1 ) and (H * 2 ). Suppose that a > 0, p > 1, q > max(1, p − 1), (23) holds, and the initial function u 0 ∈ C(R n \ S) satisfies the inequality with some constants c 0 > 0 and µ ∈ R, where γ > µ(q − p + 1) + p.
Then problem (21) has no positive solutions u in (R n \S)×[0, T ] for any arbitrarily small T > 0.
Proof. The proof of Theorem 4.2 is based on a priori estimates for u with test functions ξ 1 R (x)ψ R (x)T τ (t), where ξ R are defined as in (H * 2 ), and can be completed similarly to that of Theorem 3.2.
By an appropriate modification of the proof of Theorem 4.2, one can obtain the following result.