- Open Access
Existence of strong solutions for a class of semilinear evolution equations with nonlocal initial conditions
© Chen et al; licensee Springer. 2012
- Received: 6 April 2012
- Accepted: 12 June 2012
- Published: 12 June 2012
This paper discusses the existence of strong solutions for a class of semilinear evolution equations with nonlocal initial conditions in Hilbert spaces. The discussion is based on analytic semigroups theory and fixed point theorem. An application to a partial differential equation with nonlocal condition is also considered.
Mathematics Subject Classification(2010): 34G20; 34K30; 35D35; 47D06.
- evolution equation
- nonlocal initial condition
- strong solution
- analytic semigroups
where A: D(A) ⊂ H → H is a positive definite self-adjoint operator, f: J × H → H is given function satisfying some assumptions, J denote the real compact interval [0, a], a > 0 is a constant, 0 < t1 < t2 < · · · < t m ≤ a, m ∈ ℕ, γ i are real numbers, γ i ≠ 0, i = 1, 2, ..., m.
In 1990, Byszewski and Lakshmikantham  first investigated the nonlocal problems. They studied and obtained the existence and uniqueness of mild solutions for nonlocal differential equations. Since it is demonstrated that the nonlocal problems have better effects in applications than the classical Cauchy problems, differential equations with nonlocal conditions were studied by many authors and some basic results on nonlocal problems have been obtained, see [2–10] and the references therein. The importance of nonlocal conditions have also been discussed in [11–15]. For example, Deng  used the nonlocal condition of type (2) to describe the diffusion phenomenon of a small amount of gas in a transparent tube. In this case, condition (2) allows the additional measurements at t i , i = 1, 2, ..., m, which is more precise than the measurement just at t = 0. In , Byszewski pointed out that if γ i ≠ 0, i = 1, 2, ..., m, then the results can be applied to kinematics to determine the location evolution t → u(t) of a physical object for which we do not know the positions u(0), u(t1), ..., u(t m ), but we know that the nonlocal condition (2) holds. Consequently, to describe some physical phenomena, the nonlocal condition can be more useful than the standard initial condition.
in a reflexive Banach space X, but the conditions in [2, 3] are very strong and some of them can not be satisfied in applications. In this paper, we obtained the existence of strong solutions for the nonlocal problem (1)-(2) in a frame of abstract Hilbert spaces. Furthermore, an optimal condition (see condition (H1)) on the coefficients γ i (i = 1, 2, ..., m) to guarantee that the nonlocal problem (1)-(2) has solutions has been obtained. At last, we demonstrated that the abstract results obtained can be applied to the parabolic partial differential equation with nonlocal conditions. Our discussions are based on analytic semigroups theory and the famous Schauder's fixed point theorem.
Let H be a Hilbert space with inner product (·,·), then is the norm on H induced by inner product (·,·). We denote by C(J, H) the Banach space of all continuous H-value functions on interval J with the maximum norm and by the Banach space of all linear and bounded operators on H.
Since the positive definite self-adjoint operator A has compact resolvent, the embedding D(A) ↪ H is compact, and therefore T (t)(t ≥ 0) is also a compact semigroup.
where Γ(·) is the Euler gamma function. is injective, and A α can be defined by A α = (A-α)-1 with the domain D(A α ) = A-α(H). For α = 0, let A α = I.
We endow an inner product (·,·) α = (A α ·, A α ·) to D(A α ). Since A α is a closed linear operator, it follows that (D(A α ), (·,·) α ) is a Hilbert space. We denote by H α the Hilbert space (D(A α ), (·,·) α ). Especially, H0 = H and H1 = D(A). For 0 ≤ α < β, H β is densely embedded into H α and the embedding H β ↪ H α is compact. For the details of the properties of the fractional powers, we refer to  and .
If u0 ∈ H and h ∈ L1(J, H), the function u given by (8) belongs to C(J, H), which is known as a mild solution of the LIVP(7). If a mild solution u of the LIVP(7) belongs to W1,1(J, H) ∩ L1(J, H1) and satisfies the equation for a.e. t ∈ J, we call it a strong solution.
Moreover, u∈ W 1,2(J, H) ∩ L2(J, H1) is a strong solution of the LNP (12)-(13).
From (15) and (18), we know that u satisfies (14).
Inversely, we can verify directly that the function u ∈ C(J, H) given by (14) is a mild solution of the LNP (12)-(13).
and it is a strong solution.
We note that u(t) defined by (14) is the mild solution of the LIVP (7) for . By the representation (8) of mild solution, u(t) = T (t)u(0) + v(t), where Since the function v(t) is a mild solution of the LIVP (7) with the null initial value u(0) = θ, v has the regularity (19). By the analytic property of the semigroup T (t), T (t i )u(0) ∈ D(A) H1 / 2. Hence, . Using the regularity (19) again, we obtain that u ∈ W 1,2(J, H) ∩ L2(J, H1) and it is a strong solution of the LNP (12)-(13). □
then Ω r is a bounded closed and convex set on C(J, H).
Theorem 1 Let A be a positive definite self-adjoint operator in Hilbert space H and it have compact resolvent, f: J × H → H be continuous. If condition (H1) and the following condition
(H2) For some r > 0, there exists a function φ ∈ L(J, ℝ+) such that for all t ∈ J and u ∈ H satisfying ||u|| ≤ r, ||f(t, u) || ≤ φ(t), hold, then the problem (1)-(2) has at least one strong solution u ∈ W 1,2(J, H) ∩ L2(J, H1).
By assumption (H1) and Lemma 1, it is easy to see that the mild solution of problem (1)-(2) is equivalent to the fixed point of the operator Q. In the following, we will prove that Q has a fixed point by using the famous Schauder Fixed Point Theorem.
That is, Q is continuous on C(J, H).
Subsequently, we prove that Q: C(J, H) → C(J, H) is a compact operator. Let . By , we can prove that the operator Q defined by (20) maps C(J, H) into C ν (J, H α ). By Arzela-Ascoli's theorem, the embedding C ν (J, H α ) ↪ C(J, H) is compact. This implies that Q: C(J, H) → C(J, H) is a compact operator. Combining this with the continuity of Q on C(J, H), we know that Q: C(J, H) → C(J, H) is a completely continuous operator.
Therefore, Q(Ω R ) ⊂ Ω R . Thus, Q: Ω R → Ω R is a completely continuous operator.
By Schauder Fixed Point Theorem, we know that Q has at least one fixed point u ∈ Ω R . Since u is mild solution of the LNP (12)-(13) for h(·) = f(·, u(·)), by Lemma 1, u ∈ W 1,2(J, H) ∩ L2(J, H1) is a strong solution of the problem (1)-(2).
We have the following existence result.
which implies Q(Ω R ) ⊂ Ω R . Thus, Q: Ω R → Ω R is a completely continuous operator. By Schauder Fixed Point Theorem, we know that Q has at least one fixed point u ∈ Ω R . Since u is mild solution of the LNP (12)-(13) for h(·) = f(·, u(·)), by Lemma 1, u ∈ W 1,2(J, H) ∩ L2(J, H1) is a strong solution of the problem (1)-(2).
where M is defined by (23).
where J = [0, a], 0 < t1 < t2 < ⋯ < t m ≤ a, γ i are real numbers, γ i ≠ 0, i = 1, 2, ..., m, f: [0, 1] × J × ℝ → ℝ is continuous.
Let f(t, u(t)) = f(·, t, u(·, t)), then the problem (24) can be rewritten into the abstract form of problem (1)-(2).
Theorem 3 If the nonlinear term f(x, t, u(x, t)) = sin u(x, t)/(t1/2 + 1), x ∈ [0, 1], t ∈ J and , then the problem (24) has at least one strong solution .
Proof. Let φ(t) = t-1/2, from the condition , we easily see that the conditions (H1) and (H2) hold. Hence by Theorem 1, the problem (24) has a strong solution in the sense of L2(0, 1; ℝ). □
Theorem 4 If , f: [0, 1] × J × ℝ → ℝ is continuous and satisfies the following conditions
(P1) For some r > 0, there exists a function φ ∈ L(J, ℝ+) such that for all t ∈ J, x ∈ [0, 1] and u ∈ ℝ, | u |≤ r, | f(x, t, u(x, t)) |≤ φ(t),
for any r > 0, μ ∈ (0, 1) and (x, t, ξ), (y, s, η) ∈ [0, 1] × J × [-r, r], then the problem (24) has at least one classical solution u ∈ C2+μ,1+μ/2([0, 1] × J).
Proof. From the condition and assumption (P1), it is easy to verify that the conditions (H1) and (H2) are satisfied. Hence by Theorem 1, the problem (24) has at least one strong solution in the sense of L2(0, 1; ℝ). Since the nonlinear term f satisfies the condition (P2), by using the similar regularization method in , we can prove that u ∈ C2+μ,1+μ/2([0, 1] × J) is a classical solution of the problem (24). □
This research was supported by NNSF of China (10871160) and NNSF of China (11061031).
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