Bulletin T.CXXXIII de l’Académie serbe des sciences et des arts − 2006 Classe des Sciences mathématiques et naturelles Sciences mathématiques, N\underline{o} 31

GENERALIZED SOLUTIONS TO SINGULAR INITIAL-BOUNDARY HYPERBOLIC PROBLEMS WITH NON-LIPSHITZ NONLINEARITIES^{1}

IRINA KMIT

(Presented at the 1st Meeting, held on February 24, 2006)

A b s t r a c t. We prove the existence and uniqueness of global generalized solutions in a Colombeau algebra of generalized functions to semilinear hyperbolic systems with nonlinear boundary conditions. Our analysis covers the case of non-Lipschitz nonlinearities both in the differential equations and in the boundary conditions. We admit strong singularities in the differential equations as well as in the initial and boundary conditions.

AMS Mathematics Subject Classification (2000): 35L50, 35L67, 35D05 Key Words:

1. Introduction

We study existence and uniqueness of global generalized solutions to mixed problems for semilinear hyperbolic systems with nonlinear nonlocal

1This paper was presented at the Conference GENERALIZED FUNCTIONS 2004, Topics in PDE, Harmonic Analysis and Mathematical Physics, Novi Sad, September 22– 28, 2004

boundary conditions. Specifically, in the domain Π = {(x, t) | 0 < x < l, t > 0} we study the following problem:

(∂_{t} + Λ(x, t)∂_{x})U = F (x, t, U),  (x, t) ∈ Π  (1)
U(x, 0) = A(x),  x ∈ (0, l)  (2)
U_{i}(0, t) = H_{i}(t, V (t)), k + 1 ≤ i ≤ n,  t ∈ (0, ∞)
U_{i}(l, t) = H_{i}(t, V (t)), 1 ≤ i ≤ k,  t ∈ (0, ∞) ,  (3)

where U, F, and A are real n-vectors, Λ = diag(Λ_{1}, …, Λ_{n}) is a diagonal matrix, Λ_{1}, …, Λ_{k} < 0, Λ_{k+1}, …, Λ_{n} > 0 for some 1 ≤ k ≤ n, and V (t) = (U_{1}(0, t), …, U_{k}(0, t), U_{k+1}(l, t), …, U_{n}(l, t)). Due to the conditions imposed on Λ, the system (1) is non-strictly hyperbolic. Note also that the boundary of Π is not characteristic. We will denote H = (H_{1}, …, H_{n}).

Special cases of (1)–(3) arise in laser dynamics [7, 20, 21] and chemical kinetics [22].

All the data of the problem are allowed to be strongly singular, namely, they can be of any desired order of singularity. This entails nonlinear superpositions of distributions in the right-hand sides of (1)–(3), including compositions of the singular initial data and the singular characteristic curves. To tackle this complication, we use the framework of the Colombeau algebra of generalized functions G(\overline{Π}) [1, 16]. We show that all superpositions appearing here are well defined in G(\overline{Π}).

We establish a positive existence-uniqueness result in G(\overline{Π}) for the problem (1)–(3) with strongly singular initial data and with nonlinearities of the following type (more detailed description is given in Section 3): The functions F and H may be non-Lipschitz with less than quadratic growth in U and V.

For different aspects of the subject we refer the reader to sources [3, 4, 10, 11, 12, 14, 15, 16, 17]. The essential assumption made on F in papers [12, 16] is that grad_{U}F is globally bounded uniformly over (x, t) varying in any compact set. In contrast to [12, 16], in [11] we investigated the problem (1)– (3) with Colombeau-Lipschitz nonlinearities in (1) and (3). This means that the functions F and H are Lipschitz with Colombeau generalized numbers as Lipschitz constants and therefore their gradients are not globally bounded.

M. Nedeljkov and S. Pilipović [14, 15] deal with Cauchy problems for semilinear hyperbolic systems (1) with F slowly increasing at the infinity. The nonlinear term is replaced by a suitable regularization F_{ε} having a bounded gradient with respect to U for every fixed ε and converging to F as ε → 0. The regularized system is solved in G(ℝ^{2}). Moreover, in [14] the components of Λ are allowed to be 1-tempered generalized functions. The authors replace Λ by a regularization which is a 1-tempered generalized function of bounded growth and solve the regularized problem.

T. Gramchev [3, 4] investigates weak limits for semilinear hyperbolic systems and nonlinear superpositions for strongly singular distributions appearing in these systems. He establishes an optimal link between the singularity of the initial data and the growth of the nonlinear term. Weak limits of strongly singular Cauchy problems for semilinear hyperbolic systems with bounded, sublinear, and superlinear growth are investigated in [2, 9, 18, 19].

In the present paper we develop some results of [10] and [12] to the case of non-Lipschitz nonlinearities in (1) and (3). In Section 2 we compile some facts about Colombeau algebra of generalized functions. In Section 3 we state and prove our main result.

2. Preliminaries

In this section we summarize the relevant material on the full version of Colombeau algebras of generalized functions.

Let Ω ⊂ ℝ^{n} be a domain in ℝ^{n}. By G(Ω) and G(\overline{Ω}) we denote the full version of Colombeau algebra of generalized functions over Ω and \overline{Ω}, respectively. To define G(Ω) and G(\overline{Ω}), we first introduce the mollifier spaces used to parametrize the regularizing sequences of generalized functions. Given q ∈ ℕ_{0}, denote

\begin{matrix}A_{q}(ℝ) = \{φ ∈ D(ℝ) | ∫ φ(x) dx = 1, ∫ x^{k}φ(x) dx = 0 for 1 ≤ k ≤ q\}, \\ A_{q}(ℝ^{n}) = \{φ(x_{1}, …, x_{n}) = ∏_{i=1}^{n} φ_{0}(x_{i}) | φ_{0} ∈ A_{q}(ℝ)\}. \end{matrix}

Set

E(\overline{Ω}) = \{u : A_{0} × \overline{Ω} → ℝ | u(φ, .) ∈ C^{∞}(\overline{Ω}) ∀φ ∈ A_{0}(ℝ)\}.

We define the algebra of moderate elements E_{M}(\overline{Ω}) to be the subalgebra of E(\overline{Ω}) consisting of the elements u ∈ E(\overline{Ω}) such that

∀K ⊂ \overline{Ω} compact, ∀α ∈ ℕ_{0}^{n}, ∃N ∈ ℕ such that ∀φ ∈ A_{N}(ℝ^{n})

∃C > 0, ∃η > 0 with sup_{x∈K} |∂^{α}u(φ_{ε}, x)| ≤ Cε^{−N}, 0 < ε < η,

where φ_{ε}(x) = 1/ε^{n}φ(x/ε). The ideal N (\overline{Ω}) (see [5]) consists of all u ∈ E_{M}(\overline{Ω}) such that

∀K ⊂ \overline{Ω} compact, ∃N ∈ ℕ such that ∀q ≥ N, ∀φ ∈ A_{q}(ℝ^{n})

∃C > 0, ∃η > 0 with sup_{x∈K} |u(φ_{ε}, x)| ≤ Cε^{q−N}, 0 < ε < η.

Finally,

G(\overline{Ω}) = E_{M}(\overline{Ω})/N (\overline{Ω}).

This is an associative and commutative differential algebra. The algebra G(Ω) on an open set Ω is constructed in the same manner, with Ω in place of \overline{Ω}. Note that G(Ω) admits a canonical embedding of D′(Ω). We will use the notation U = [(u(φ, x))_{φ∈A_{0}(ℝ^{n})}] for the elements U of G(Ω) with the representative u(φ, x).

One of the advantages of using the Colombeau algebra of generalized functions G lies in the fact that in a variety of important cases the division by generalized functions, in particular the division by discontinuous functions and measures, is defined in G. Complete description of the cases when the division is possible in the full version of Colombeau algebras is given by the following criterion of invertibility [10] (the criterion of invertibility for the special version of Colombeau algebras G_{s}(Ω) is proved in [6]):

Theorem 1. Let U ∈ G(Ω) (resp. U ∈ G(\overline{Ω})). Then the following two conditions are equivalent:

(i) U is invertible in G(Ω) (resp. in G(\overline{Ω})), i.e., there exists V ∈ G(Ω) (resp. V ∈ G(\overline{Ω})) such that UV = 1 in G(Ω) (resp. in G(\overline{Ω})). (ii) For each representative (u(φ, x))_{φ∈A_{0}(ℝ^{n})} of U and each compact set K ⊂ Ω (resp. K ⊂ \overline{Ω}) there exists p ∈ ℕ such that for all φ ∈ A_{p}(ℝ^{n}) there is η > 0 with inf_{K} |u(φ_{ε}, x)| ≥ ε^{p} for all 0 < ε < η.

3. Existence and uniqueness of a Colombeau generalized solution

We will need a notion of a generalized function whose growth is more restrictive than the 1/ε-growth (as in the definition of E_{M}).

Definition 2. ([10]) Let Ω ⊂ ℝ^{n} be a domain in ℝ^{n}. Given a function γ : (0, 1) ↦ (0, ∞), we say that an element U ∈ G(Ω) (resp. U ∈ G(\overline{Ω})) is locally of γ-growth, if it has a representative u ∈ E_{M}(Ω) (resp. u ∈ E_{M}(\overline{Ω})) with the following property:

For every compact set K ⊂ Ω (resp. K ⊂ \overline{Ω}) there is N ∈ ℕ such that for every φ ∈ A_{N}(ℝ^{n}) there exist C > 0 and η > 0 with sup_{x∈K} |u(φ_{ε}, x)| ≤ Cγ^{N}(ε)

for 0 < ε < η.

Let K ⊂ ℝ^{m} be a compact. Let U(x, y) and V (x, y), as functions of x, are in G(K) for each y ∈ ℝ^{n}. We will say that U is bounded by V and write U ≤ V if U and V have representatives u(·, y) ∈ E_{M}(K) and v(·, y) ∈ E_{M}(K), respectively, satisfying the following property for some N ∈ ℕ: For every φ ∈ A_{N}(ℝ^{n}) there exists η > 0 such that |u(φ_{ε}, x, y)| ≤ v(φ_{ε}, x, y) for all x ∈ K, y ∈ ℝ^{n}, and 0 < ε < η.

We will write F (x, y) ∈ C_{y}^{∞}(ℝ^{n}; G(\overline{Π})) if F is C^{∞} with respect to y ∈ ℝ^{n} and ∂_{y}^{α}F (·, y) ∈ G(\overline{Π}) for every α ∈ ℕ_{0}^{n} and each y ∈ ℝ^{n}. Here ∂_{y}^{α} = \frac{∂^{α_{1}+...+αn}}{∂_{y_{1}}^{α_{1}}...∂_{yn}^{αn}}.

We now make assumptions on the initial data of the problem (1)–(3). Let γ(ε) be a function from (0, 1) to (0, ∞) such that

γ(ε)^{γ^{N} (ε)} = O(\frac{1}{ε})  (4)

for each N ∈ ℕ. Assume that

1. Λ(x, t) ∈ (G(\overline{Π}))^{n}, A(x) ∈ (G[0, l])^{n}.

2. Λ_{i} for i ≤ n are locally of γ-growth on \overline{Π} and invertible on \overline{Π} (see Theorem 1).

3. ∂_{x}Λ_{i} for i ≤ n are locally of γ-growth on \overline{Π}.

4. F (x, t, y) ∈ (C_{y}^{∞}(ℝ^{n}; G(\overline{Π}))^{n}, H(t, z) ∈ (C_{z}^{∞}(ℝ^{n}; G[0, ∞)))^{n}.

5. For every compact set K ⊂ \overline{Π}, i ≤ n, and α ∈ ℕ_{0}^{n+2}, the function D^{α}F_{i}(x, t, y) is bounded by a polynomial in G(K)[y] (polynomials over y with coefficients in G(K)).

6. For every compact set K ⊂ [0, ∞), i ≤ n, and α ∈ ℕ_{0}^{n+1}, the function D^{α}H_{i}(t, z) is bounded by a polynomial in G(K)[z].

7. supp A_{i}(x) ⊂ (0, l) and supp H_{i}(t, 0) ⊂ (0, ∞) for i ≤ n.

Assumptions imposed on Λ_{i} allow them to be strongly singular and, even more, to have any desired order of singularity. Assumptions 4–6 state that, given U ∈ (G(\overline{Π}))^{n} and V ∈ (G[0, ∞))^{n}, F (x, t, U) and H(t, V) are well defined in the Colombeau algebra G. The last assumption ensures the compatibility of (2) and (3) of any desired order.

Given T > 0, denote

Π^{T} = {(x, t) | 0 < x < l, 0 < t < T }.

To state the main result of the paper, we suppose additionally that at least one of the following two assumptions holds.

Assumption 8.

a) H(t, V) is smooth in t, V and the mapping V ↦ ∇_{V} H(t, V) is globally bounded, uniformly over t varying in compact subsets of [0, ∞);

b) Given T > 0, there exists C_{F} such that for all 1 ≤ i ≤ n we have

|∇_{y}F_{i}(x, t, y)| ≤ C_{F} log log D(x, t, y),

where D(x, t, y) ∈ G(\overline{Π}^{T})[y].

Assumption 9.

a) Given T > 0, there exists C_{H} such that for all 1 ≤ i ≤ n we have

|∇_{z}H_{i}(t, z)| ≤ C_{H}(log log B(t, z))^{1/4},

where B(t, z) ∈ G[0, T][z].

b) Assumptions 2 and 3 are true with γ(ε) = O((log log 1/ε)^{1/4});

c) Given T > 0, there exists C_{F} such that for all 1 ≤ i ≤ n we have

|∇_{y}F_{i}(x, t, y)| ≤ C_{F} (log log D(x, t, y))^{1/4},

where D(x, t, y) ∈ G(\overline{Π}^{T})[y].

Theorem 3. Assume that Assumption 8 or 9 is true. Under Assumptions 1–7 where the function γ is specified by (4), the problem (1)–(3) has a unique solution U ∈ G(\overline{Π}).

Set

E_{U}(α_{1}, α_{2}; T) = max\{|∂_{x}^{α_{1}}∂_{t}^{α_{2}}U_{i}(x, t)| | (x, t) ∈ \overline{Π}^{T} , 1 ≤ i ≤ n\}, E_{F} (α_{1}, α_{2})

= max\{|∂_{x}^{α_{1}}∂_{t}^{α_{2}}F_{i}(x, t, y)| | (x, t, y) ∈ \overline{Π}^{T} ×\{y : |y| ≤ E_{U}(0, 0; T)\}, 1 ≤ i ≤ n\}, E_{H}(α) = max{|∂_{t}^{α}H_{i}(t, z)| | (t, z) ∈ [0, T]×{z : |z| ≤ E_{U}(0, 0; T)}, 1 ≤ i ≤ n},

L_{∇F} (U) = max\{|∇_{U}F_{i}(x, t, U(x, t))| : (x, t) ∈ \overline{Π}^{T} , 1 ≤ i ≤ n\},
L_{∇H}(V) = max{|∇_{V} H_{i}(t, V (t))| : t ∈ [0, T], 1 ≤ i ≤ n}.

Simplifying the notation, we drop the dependence of E_{F} (α_{1}, α_{2}), E_{H}(α), L_{∇F} (U) and L_{∇H}(V) on T. Note that T will be a fixed positive number.

To prove the theorem, we need the following lemma.

Lemma 4. Assume that the initial data Λ, F, A, and H are smooth with respect to all their arguments and satisfy Assumption 7, ∇_{y}F (x, t, y) is bounded on K × ℝ^{n} for every compact K ⊂ \overline{Π}, and ∇_{z}H(t, z) is bounded on K × ℝ^{n} for every compact K ⊂ [0, ∞). Then, given T > 0, the problem (1)– (3) has a unique smooth solution U in \overline{Π}^{T} satisfying the following a priori estimates:

E_{U}(0, 0; T) ≤ P_{1,0}(\frac{1}{1 − q_{0}t_{0}} , n, L_{∇H}(V))

×P_{2,0}( max_{x∈[0,l],1≤i≤n} |A_{i}(x)|, max_{(x,t)∈\overline{Π}^{T} ,1≤i≤n} |F_{i}(x, t, 0)|, max_{t∈[0,T],1≤i≤n} |H_{i}(t, 0)|)

(5)

and

E_{U}(m, 0; T) ≤ P_{1,m}(\frac{1}{1 − q_{m}t_{m}} , n, L_{∇H}(V))

×P_{2,m}(n, max_{x∈[0,l],1≤i≤n} |A_{i}^{(m)}(x)|, max_{0≤α_{1}+α_{2}≤m−1} E_{Λ^{−1}}(α_{1}, α_{2}; T),

max_{0≤α_{1}+α_{2}≤m} E_{Λ}(α_{1}, α_{2}; T), max_{1≤|β|+α_{1}+α_{2}≤m} E_{∂_{U}^{β} F} (α_{1}, α_{2}), max_{1≤|β|+α_{1}≤m} E_{∂_{V}^{β} H}(α_{1}),

L_{∇F} (U), L_{∇H}(V), max_{1≤α_{1}+α_{2}≤m−1} E_{U}(α_{1}, α_{2}; T)), m ∈ ℕ,

(6)

where

q_{m} = (nL_{∇F} (U) + mE_{Λ}(1, 0; T))(1 + nL_{∇H}(V)), m ∈ ℕ_{0},
t_{m} ≤ min{L/E_{Λ}(0, 0; T), 1/q_{m}}, m ∈ ℕ_{0},

P_{1,m} is a polynomial of degree 3⌈T/t_{m}⌉ with all coefficients identically equal to 1, P_{2,m} is a polynomial whose degree depends on m but neither on T nor on t_{m} and whose coefficients are positive constants depending only on m and T.

The lemma directly follows from the proof of Theorem 2.1 in [11]. Note that similar global a priori estimates for E_{U}(α_{1}, α_{2}; T), where α_{1} + α_{2} ≤ m, follow from the estimates (5) and (6) as well as from the system (1) and its suitable differentiations.

P r o o f o f t h e t h e o r e m. The classical smooth solution to the problem (1)–(3) satisfying estimates (5) and (6) in \overline{Π}^{T} for any m ∈ ℕ_{0} and T > 0 can be constructed by the sequential approximation method. We now use this solution to construct a representative of the Colombeau solution. According to the assumptions of the theorem, we consider all the initial data as elements of the corresponding Colombeau algebras. We choose representatives λ, a, f, and h of Λ, A, F, and H, respectively, with the properties required in the theorem. Let ϕ = φ ⊗ φ ∈ A_{0}(ℝ^{2}). Consider a prospective representative u = u(ϕ, x, t) of U which is the classical smooth solution to the problem (1)–(3) with the initial data λ(ϕ, x, t), a(φ, x), f(ϕ, x, t, u(ϕ, x, t)), h(φ, t, v(φ, t)), where v(φ, t) = (u_{1}(ϕ, 0, t), …, u_{k}(ϕ, 0, t), u_{k+1}(ϕ, l, t), …, u_{n}(ϕ, l, t)). For the existence part of the proof, we have to show that u ∈ E_{M}, i.e., to obtain moderate growth estimates of u(ϕ_{ε}, x, t) in \overline{Π}^{T} for any T > 0 in terms of the regularization parameter ε. Set f_{ε}(x, t, y) = f(ϕ_{ε}, x, t, y), h_{ε}(t, z) = h(φ_{ε}, t, z), and λ_{ε}(x, t) = λ(ϕ_{ε}, x, t).

In the proof we will use a modified notion of E_{M}(\overline{Π}). Namely, let u ∈ E_{M}(\overline{Π}) iff u ∈ E(\overline{Π}) and for every compact set K ⊂ \overline{Π} there is N ∈ ℕ such that for every φ ∈ A_{N}(ℝ^{n}) there exists η > 0 with sup_{x∈K} |u(ϕ_{ε}, x, t)| ≤ γ^{N}(ε)

for all 0 < ε < η.

Fix an arbitrary T > 0. Fix N ∈ ℕ to be so large that for all φ ∈ A_{N}(ℝ) there exists ε(φ) such that for all ε < ε(φ) the following conditions are true:

(a) The moderate estimate (see the definition of E_{M}) holds for a(φ_{ε}, x), f(ϕ_{ε}, x, t, 0), and h(φ_{ε}, t, 0).

(b) The invertibility estimate (see Theorem 1) holds for λ(ϕ_{ε}, x, t).

(c) The local-γ-growth estimate (see Definition 2) holds for λ(ϕ_{ε}, x, t) and ∂_{x}λ(ϕ_{ε}, x, t).

(d) For all i ≤ n, all y, z ∈ ℝ^{n}, and some C > 0 the following estimates are true: |∇_{y}f_{i,ε}(x, t, y)| ≤ C_{F} log log d(ϕ_{ε}, x, t, y) and |∇_{z}h_{i,ε}(t, z)| ≤ C if Assumption 8 is fulfilled or |∇_{y}f_{i,ε}(x, t, y)| ≤ C_{F} sup_{(x,t)∈\overline{Π}^{T}}(log log d(ϕ_{ε}, x, t, y))^{1/4}

and |∇_{z}h_{i,ε}(t, z)| ≤ C_{F} sup_{(x,t)∈\overline{Π}^{T}} (log log b(φ_{ε}, t, z))^{1/4} if Assumption 9 is ful-

filled, where b and d are representatives of B and D, respectively. (e)The moderate estimate holds for the coefficients of the polynomial d(ϕ_{ε}, x, t, y) if Assumption 8 is fulfilled or for the coefficients of the polynomials d(ϕ_{ε}, x, t, y) and b(φ_{ε}, t, z) if Assumption 9 is fulfilled.

Given φ ∈ A_{N}(ℝ), denote by p_{1,m}(φ), p_{2,m}(φ), q_{m}(φ), and t_{m}(φ) the value of, respectively, P_{1,m}, P_{2,m}, q_{m}, and t_{m}, where U(x, t), Λ(x, t), A(x), F (x, t, U(x, t)), H(t, V (t)), L_{∇F} (U), L_{∇H}(V) are replaced by their representatives u(ϕ, x, t), λ(ϕ, x, t), a(φ, x), f(ϕ, x, t, u(ϕ, x, t)), h(φ, t, v(φ, t)), L_{∇f}(u), and L_{∇h}(v), respectively. On the account of (5) and (6), it suffices to prove the moderate estimates for p_{1,m}(φ_{ε}) and p_{2,m}(φ_{ε}) for all m ∈ ℕ_{0}.

From the condition (a) and the description of P_{2,0} given in Lemma 4 it follows that [(p_{2,0}(φ))_{φ∈A_{0}(ℝ)}] is a Colombeau generalized number and hence has the moderateness property. This means that there exists N_{1} ≤ N such that for all φ ∈ A_{N_{1}}(ℝ) there is 0 < η(φ) < ε(φ) with

|p_{2,0}(φ_{ε})| ≤ ε^{−N_{1}}, 0 < ε < η(φ).  (7)

Note that any U ∈ G(\overline{Π}) has the following property: there exists N_{2} ∈ ℕ such that for all φ ∈ A_{N_{1}+N_{2}}(ℝ) there is ε_{0}(φ) ≤ η(φ), where the value of η(φ) is the same as in (7), with

sup_{\overline{Π}^{T}} |u(ϕ_{ε}, x, t)| ≤ ε^{−N_{1}−N_{2}}, 0 < ε < ε_{0}(φ),  (8)

with the constant N_{1} being the same as in (7). Obviously, any increase of N_{2} and any decrease of ε_{0}(φ) will keep this property true. This will allow us to adjust the values of N_{2} and ε_{0}(φ) according to our purposes.

Set u_{ε}(x, t) = u(ϕ_{ε}, x, t) and v_{ε}(t) = (u_{1,ε}(0, t), …, u_{k,ε}(0, t), u_{k+1,ε}(l, t), …, u_{n,ε}(l, t)). Given φ ∈ A_{N_{1}+N_{2}}(ℝ), let us consider the estimates (5) and (6) with p_{1,m}(φ_{ε}) and p_{2,m}(φ_{ε}) in place of P_{1,m} and P_{2,m}, respectively, where 0 < ε < η(φ) and the value of η(φ) is the same as in (7). On the account of these estimates, we will obtain the existence once we prove the following assertion:

(ι) given m ∈ ℕ_{0}, a positive integer N(m), where N(0) = N_{2}, can be chosen so that for all φ ∈ A_{N_{1}+N_{2}}(ℝ) there exists ε_{m}(φ) such that

[2n(1+L_{∇h_{ε}}(v_{ε}))]^{6T (1+L_{∇hε}(v_{ε}))(L_{∇fε}(u_{ε})+mE_{λε}(1,0;T))+2T/lE_{λε}(0,0;T)+1}≤ε^{−N(m)}

(9)

for all 0 < ε < ε_{m}(φ), provided u(ϕ, x, t) ∈ E(\overline{Π}^{T}) satisfies the inequality (8).

Indeed, take t_{m}(φ_{ε}) = 1/2 min{L/E_{λ_{ε}}(0, 0; T), 1/q_{m}(φ_{ε})}. When the assertion (ι) is fulfilled, then the moderate estimates for p_{1,m}(φ_{ε}) follow from the fact that the left-hand side of (9) is an upper bound for p_{1,m}(φ_{ε}). By (5), (7), and (9) for m = 0, we have the zero-order moderate estimate (8) for u_{ε}(x, t). Moderate estimates of order m ≥ 1 for u_{ε}(x, t) are easy to obtain, using induction on m, estimates (6) and (9), assumptions imposed on the initial data, and the fact that p_{1,m}(φ_{ε}) are polynomials whose degree do not depend on ε (see Lemma 4).

Let us prove Assertion (ι) under Assumption 8. Recall that at this point N_{2} is a constant whose exact value will be fixed below. Fix φ ∈ A_{N_{1}+N_{2}}(ℝ). By (8) and Assumption 8, there exists N_{3} ∈ ℕ for which the estimate

L_{∇f_{ε}}(u_{ε}) ≤ C_{F} log log d(ϕ_{ε}, x, t, u_{ε}) ≤ C_{F} log log ε^{−N_{3}}, 0 < ε < ε_{0}(φ),

is true. Furthermore, there exist constants C > 0 and k_{i}(m) ∈ ℕ such that the left hand side of (9) is bounded from above by

C^{k_{1}(m)(log log ε^{−N_{3}}+γ^{N+1}(ε))} ≤ e^{k_{2}(m) log log ε^{−N_{3}}} γ(ε)^{k_{1}(m)γ^{N+1}(ε)}

≤ e^{log(log ε^{−N_{3}})^{k_{2}(m)}}ε^{−k_{3}(m)} ≤ (log ε^{−N_{3}})^{k_{2}(m)}ε^{−k_{3}(m)} ≤ N_{3}^{k_{2}(m)}ε^{−k_{2}(m)−k_{3}(m)},

where 0 < ε < ε_{m}(φ). Set N(m) = 2k_{2}(m) + k_{3}(m) and ε_{m}(φ) = min{η(φ), N_{3}^{−k_{2}(m)}}. It is important to note that the k_{2}(0) and k_{3}(0) can be fixed so that the estimates with m = 0 hold for all N_{2} and all φ. This makes the values N_{2} = N(0) and ε_{0}(φ), which we just fixed, well defined. Assertion (ι) now follows from the fact that φ is an arbitrary function in A_{N_{1}+N_{2}}(ℝ).

Let us now prove Assertion (ι) under Assumption 9. Following the same scheme as above, fix φ ∈ A_{N_{1}+N_{2}}(ℝ), where N_{2} will be specified below. By (8) and Assumption 9, there exist N_{3}, N_{4} ∈ ℕ such that the following estimates are true:

L_{∇f_{ε}}(u_{ε}) ≤ C_{F} log log d(ϕ_{ε}, x, t, u_{ε}) ≤ C_{F} log log(ε^{−N_{3}}), 0 < ε < ε_{0}(φ),

L_{∇h_{ε}}(v_{ε}) ≤ C_{H} log log b(φ_{ε}, t, v_{ε}) ≤ C_{H} log log(ε^{−N_{4}}), 0 < ε < ε_{0}(φ).

Furthermore, there exist C > 0 and k(m) ∈ ℕ such that the left hand side of (9) is bounded from above by

[C log log(ε^{−N_{4}})]^{1/2k(m)(log log ε^{−N_{3}−N_{4}})^{1/2}}
≤ exp{k(m) log(log(log(ε^{−N_{4}}))^{C})^{1/2}(log log ε^{−N_{3}−N_{4}})^{1/2}}
≤ exp{k(m) log(log ε^{−N_{3}−N_{4}})^{C}} = (log ε^{−N_{3}−N_{4}})^{Ck(m)}

= ((N_{3} + N_{4}) log ε^{−1})^{Ck(m)} ≤ (N_{3} + N_{4})^{⌈Ck(m)⌉}ε^{−⌈Ck(m)⌉},

where 0 < ε < ε_{m}(φ). We now set N_{2} = 2⌈Ck(m)⌉ and ε_{m}(φ) = min{η(φ), (N_{3}+N_{4})^{−⌈Ck(m)⌉}}. Note that C and k(0) can be fixed so that the estimates with m = 0 hold for all N_{2} and all φ. This makes the values N_{2} = N(0) and ε_{0}(φ) well defined. Assertion (ι) now follows from the fact that φ is an arbitrary function in A_{N_{1}+N_{2}}(ℝ).

Since T > 0 is arbitrary, the existence part of the proof is complete.

The proof of the uniqueness part follows the same scheme. The only difference is that now we consider the problem with respect to the difference U − W of two Colombeau solutions U and W. We hence have the problem (1)–(3) with the right hand sides M_{2},

∫_{0}^{1} ∇_{U}F (x, t, σU + (1 − σ)W) dσ · (U − W) + M_{1},

and

∫_{0}^{1} ∇_{V} H(t, σV + (1 − σ)V_{W}) dσ · (V − V_{W}) + M_{3},

in (2), (1), (3), respectively. Here M_{i} ∈ N and V_{W} are equal to V if we replace U by W. We apply the estimate (5) to the difference U − W. From the existence part of the proof we see that the first factor in the righthand side of (5) has the moderateness property. Since the second factor is negligible, the uniqueness follows. □

Example 5. Let n = 1 and

F (x, t, U) = (1+A^{2}(x, t)+B^{2}(x, t)U^{2})^{1/2} log log (1 + C^{2}(x, t) + D^{2}(x, t)U^{2})^{1/2},

where A, B, C, D ∈ G(\overline{Π}). Then

∂_{U}F (x, t, U) = \frac{1 + B^{2}U}{(1 + A^{2} + B^{2}U^{2})^{1/2}} log log (1 + C^{2} + D^{2}U^{2})^{1/2}

+\frac{D^{2}U(1 + A^{2} + B^{2}U^{2})^{1/2}}{log (1 + C^{2} + D^{2}U^{2})^{1/2}(1 + C^{2} + D^{2}U^{2})}.

The function F (x, t, U) is non-Lipschitz and satisfies Assumption 8(b).

Remark 6. Theorem 3 shows that, whatsoever singularity of the initial data of our problem and whatsoever nonlinearities of F and H allowed by Assumption 8 (or 9), the problem (1)–(3) has a unique solution in the Colombeau algebra G(\overline{Π}).

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Institute for Applied Problems of Mechanics and Mathematics Ukrainian Academy of Sciences Naukova St. 3b 79060 Lviv Ukraine

E-mail: kmit@ov.litech.net