The initial-boundary value problem for a class of third order pseudoparabolic equations

In this paper, a priori estimate for a linear third pseudoparabolic operator with bound is established, and applying the above result, the existence and uniqueness theorem of solutions for a class of nonlinear pseudoparabolic equations is obtained with the help of the homeomorphism method and the initial value method. Furthermore, an existence and uniqueness theorem of the semilinear equation is obtained as a corollary.


Introduction
Consider the initial-boundary value problem where Ω is a connected bounded subset of n-dimensional space, the boundary of Ω is piecewise smooth and has nonnegative mean curvature everywhere, D = Ω × [0, T] belongs to the Hilbert space W 2,1 2 (D), and a i,j , b i are bounded measurable functions. Using a continuous method, Sigillito [1] explored the solution for the heat equation. Elcart and Sigillito [2,3] had derived an explicit coercivity inequality and discussed the convergence of the algorithm for a semilinear third order pseudoparabolic equation of the following type: since then, there have been some further studies of other forms of parabolic equations, most of the results have focused on the discussion of algorithms [4].
In 2004, Bouziani [5,6] had derived an explicit coercivity inequality and given a sufficient condition for the existence and uniqueness of a solution to the first order parabolic equation.
Motivated by the spirit of this work and results by Brown and Lin [7], the explicit coercivity inequalities of a linear third pseudoparabolic operator with bound are obtained in Sect. 3. By using these estimates, we shall utilize the homeomorphism method and the initial value method to give a new set of sufficient conditions for the existence and uniqueness of the third order pseudoparabolic equation in this paper, which can be found in Sect. 4.

Preliminaries and lemmas
In this section, we will state some lemmas which are useful to our results.
Firstly, we will give sufficient conditions for f to be a global homeomorphism of D onto Y .
where q : [0, 1] → Y is any line in Y , there is a sequence {t n } such that t n → a, n → ∞ and lim n→∞ r(t n ) exists and is in D.
In the following for convenience, with no loss of generality, for the function q one may assume that q(t) = (1t)f (x 0 ) + ty, t ∈ [0, 1], for arbitrary x 0 ∈ D and y ∈ Y . Secondly, the comparison theorem plays an important role to prove the sufficient condition for the existence of a unique solution of the problem (1).
Definition 2.2 ([9]) Let y(t) be a solution of the scalar differential equation (4) on [t 0 , t 0 + a), then y(t) is said to be a maximal solution of (4) if, for every solution u(t) of (4) existing on [t 0 , t 0 + a), we have the inequality holds.
Then there exist a maximal solution and a minimal solution of (4) on [t 0 , t 0 + a], where α = min(a, b \ 2(M + b)). [9]) In the setting of the above, suppose that [t 0 , t 0 + b) is the largest interval in which the maximal solution y(t) of (4) exists. Let

Theorem 2.3 (Comparison theorem
and for a fixed Dini derivative where T denotes an almost countable subset of t ∈ [t 0 , t 0 + b).

The coercivity inequality
Let W 0 (D) denote the Hilbert space with the norm here |D 2 u| 2 represents the sum of the squares of all the second derivatives with respect to space variables. In this section we derive a coercivity inequality, for the pseudoparabolic operator defined by The norm | · | on W 0 is defined by where · 2 is the norm on W 2 2 (Ω), · is the norm on L 2 (D), a : W 0 (D) → L 2 (D) is continuous and a bounded function on t, x 1 , . . . , x n , u.
We assume that a ij is a symmetric matrix of measurable functions satisfying the inequality τ 2 |ξ | ≤ a i,j ξ i ξ j for some positive constant τ , all n-dimensional vectors ξ and all x in D. We also assume that the functions a ij are sufficiently regular to ensure the validity of the identity Using the inequality and the inequality is valid for u in W 0 . The next two lemmas are obtained from (5) by evident choices of ε and α. In order to facilitate statements to be made below, we define S = sup |b i -(a i,j ) x j |, a 0 = inf D a(x, t).

Lemma 3.1 The inequality
is valid for u in W 0 .

Lemma 3.2 The inequality
is valid for u in W 0 .
We define and from [2] we have the inequality Remark 3.1 Results analogous in the present situation are in Lemma 3.3.

Lemma 3.3 The inequality
is valid for u in W 0 .
The inequalities (10) and (11) imply that Denote and by further application of the arithmetic-geometric mean inequality to Combining (8)

The coercivity inequality
Denote then M is a linear operator from W 0 (D) to L 2 (D). Now let us turn our attention to the following operator equation: For all u, φ ∈ W 0 (D), we have If inf Ω f u > S 2 4τ 2 , then zero is not an eigenvalue of Mφ -f u (x, u(x))φ, so for every u ∈ W 0 (D), the operator A (u) = Mf u I is invertible and A is a local homeomorphism from W 0 (D) onto L 2 (D), where I denotes the identical operator. Furthermore, an upper bound for for positive constant α, β. Denote We may express the first line of (1) in the form We can state and prove our main theorem.
is defined on [0, a] and there exists a sequence t n → a as n → ∞ such that lim n→∞ y(t n ) = y is finite; (3) F is continuously differentiable and Proof Firstly, we prove [Mf u (u) + F u (u)] -1 ≤ δ( u ). For u, v ∈ D, it is obvious that F is continuously Frechet differentiable with It follows from the above assumption and (16) that Now Let Q : W 0 (D) → W 0 (D) be defined by by (14) and (16), we have So I + Q is invertible with and so It implies that P is invertible at every u ∈ W 0 (D), hence, P is a local homeomorphism of W 0 (D). Secondly, in view of Theorem 2.1, we need only show that P has the property (C) for any continuous function q : [0, 1] → L 2 (D). For a given y ∈ L 2 (D) and an arbitrary x 0 ∈ W 0 (D), let q(t) = (1t)y + tP(x 0 ), suppose that there exists a continuous function r : [0, a) → D ⊆ W 0 (D) such that P r(t) = q(t), t ∈ [0, a), for 0 < a ≤ 1.
Now we need to prove that there exists a real sequence {t n } such that t n → a, n → ∞ and lim n→∞ r(t n ) = r exists and is in W 0 (D). It is clear that r is differentiable in this case. We have from (18) ⎧ ⎨ ⎩ r (t) = -P (r(t)) -1 (P(x 0 ) -P(r(t))), t ∈ [0, a), Denote by D r(t) the Dini derivative of r(t) and set μ = P(r(t)) -P(x 0 ), and we have D r(t) ≤ r (t) = P r(t) -1 P(x) -P(x 0 ) ≤ μδ r(t) .