Nonexistence of Global Solutions to an Elliptic Equation with a Dynamical Boundary Condition

We consider the equation ∆u = 0 posed in Q := (0, +∞) × Ω, Ω := {x = (x′, xN )/x′ ∈ RN−1, xN > 0}, with the dynamical boundary condition B(t, x′, 0)utt + A(t, x′, 0)ut − uxN ≥ D(t, x′, 0)|u|q on Σ := (0,∞) × RN−1 × {0} and give conditions on the coefficient functions A(t, x′, 0), B(t, x′, 0) and D(t, x′, 0) for the nonexistence of global solutions.

Before discribing our result in detail, let us dwell on some literature related to equations with dynamical boundary conditions.These kind of problems have been studied for a long time; see [9], [8], [11], [5].More information is contained in the book by Lions [12]; In chapter 11 of [12], the existence of weak solutions to the Laplace equation with various nonlinear dynamical boundary conditions of parabolic and hyperbolic type is studied.More recently, Kirane [10] considered blow up for three equations with dynamical boundary conditions of parabolic and 2000 Mathematics Subject Classification: 35J99, 35L20 hyperbolic types.Later, Escher [6] addressed the questions of local solvability and blow up for such problems.Andreucci and Gianni [3] discussed the global existence and blow up issues for a degenerate parabolic problem with nonlocal dynamical boundary conditions, this on one hand.On the other hand, Apushkinskaya and Nazarov [1] present a survey on the recent results on boundary value problems with boundary conditions described by second-order Venttsel operators.They paid a special attention to nonlinear problems for elliptic and parabolic equations.They stated a priori estimates and existence results in Sobolev and Hölder spaces.
In [2], Amann and Fila derived a Fijita's type result for the Laplace equation with a parabolic dynamical boundary condition with constant coefficients posed in a half-space; they were followed by Fila and Quittner [7] who discussed the same problem as in [2] but in a bounded domain.In this paper, we generalize the results of [2] concerning blow up to inequalities with non constant coefficients rather than equations with constant coefficients.Observe also that our technique of proof is different of that used by Amann and Fila which parallels Fujita's one; we rather follow an idea from the papers of Baras and Pierre [4], Mitidieri and Pohozaev [13] which is based on a judicious choice of the test function in the weak formulation of the problem, and a scaling of the variables.

Preliminaries
The coefficient functions A(t, x , 0), B(t, x , 0) and D(t, x , 0) appearing in the boundary condition are assumed to verify the hypotheses H1.
H5.The function D is strictly positive for x outside a large ball.
Definition.By a solution of (EI) on Q subject to the conditions (1), we mean a for any test function φ ∈ C 2 0 (R + × Ω), φ(T, x , x N ) = 0 for T large enough.
For later use, we set

The Result
Now, we are in force to announce our main result.

Then problem (EI)-(1) doesn't admit global non trivial solutions.
Proof.The proof is by contradiction.So, we assume that the solution is global.Let ϕ 0 ∈ C 2 0 (R), ϕ 0 ≥ 0, ϕ 0 decreasing be such that Next, let's define ψ = ψ(t, x) as the solution of We choose ϕ(t, x ) = ϕ λ 0 (ξ), dy, where σ N /N is the volume of the unit ball; so, we have Multiplying Equation (EI) 1 by ψ and integrating, we obtain which, in the light of the Green formula, yields because on Σ, ψ = ϕ.Now, using ϕ in (2) as a test function and splitting the expression into terms, we obtain

Also, we have
So if we assume u ≥ 0, we may then write In the case where B tt − A t = 0, the solution u may change sign.By (H1) and (H2), (3) becomes Using the ε-Young inequality to estimate the right hand side of (4), we obtain for some positive constants ε and C(ε), and where p + q = pq.
For ε small enough, we obtain The right hand side of ( 5) is finite thanks to our choice of the test function.
admits, for each φ bounded and uniformly continuous on R N −1 , a unique maximal solution u φ .Moreover, they showed that -If T φ < ∞ then lim -If q ≤ N/(N − 1) then every nonzero maximal solution blows up in finite time.
-If q > N/(N −1) then there are solutions that exist globally as well as solutions that blow up in finite time.
In this case, (8) reads So if we choose χ = 1, then q ≤ N/(N − 1) for N ≥ 2; this corresponds to the result of Amann and Fila as expected.
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θ a positive real and λ any real greater than p, such that the integrals