Regularity of Weak Solutions of Semilinear Parabolic Systems of Arbitrary Order

Hans-Christoph Grunau Wolf von Wahl

Abstract Let u be a weak solution of the initial boundary value problem for the semilinear parabolic system of order 2m : u′(t) + Au(t) + f(t, ., u, ..., ∇^{m}u) = 0. Let f satisfy controllable growth conditions. Then u is smooth.

This result is proved by a kind of continuity method, where the time t is the parameter of continuity.

Classification: 35D10, 35K60, 35K50.

1. Introduction

We are interested in the regularity problem for weak solutions u to the initial boundary value problem for systems

u′(t) + A(t)u + f(t, ., u, ..., ∇^{m}u) = 0, t ∈ [0, T]. (1)

A(t) is a positive uniformly elliptic operator of order 2m in a bounded domain Ω ⊂ IR^{n}. The data (e.g. the initial value, the function f, the coefficient functions of A, etc.) are assumed to be sufficiently smooth.

A weak solution u is understood to be in the space L^{2}((0, T), H_{0}^{m,2}(Ω)) ∩ L^{∞}((0, T), L^{2}(Ω)). On f we impose the so called controllable growth conditions

|f(t, ., u, ..., ∇^{m}u)| ≤ c(1 + ∑_{ν=0}^{m} |∇^{ν}u|^{\frac{n+4m}{n+2ν}}). (2)

These are the weakest growth conditions under which a weak solution as above can be dealt with by using testing functions χ with χ′ ∈ L^{2}((0, T), L^{2}(Ω)), χ ∈ L^{2}((0, T), H_{0}^{m,2}(Ω)).

In [Wa1] single equations were studied. Additionally the sign condition

u · f(t, ., u, ..., ∇^{m}u) ≥ c′u − c′′u^{2}, (3)

was imposed. It was shown that any weak solution is regular. Moreover the

condition (3) enabled the author to give a-priori estimates for ∫_{0}^{T} ∥u′∥_{p}^{p} ds +

∫_{0}^{T} ∥u∥_{2m,p}^{p} ds, if p > n+1. In some sense, [Wa1] extends a result of Ladyženska-

ja, Solonnikov and Ural’ceva [LSU, V.2] concerning quasilinear second order equations to arbitrary order.

In the case of elliptic systems, Luckhaus [Lu] discussed the regularity problem for weak solutions completely separately from the problem of finding a-priori estimates. He showed that any weak solution of a semilinear elliptic system is regular if (beside some smoothness assumptions) only controllable growth conditions are imposed on the nonlinear part.

Because of the far reaching analogy between elliptic and parabolic problems it seems reasonable to conjecture that regularity will still hold if the sign condition (3) is omitted, i.e. if no a-priori energy bound is available.

The works of Campanato [Ca] and Marino, Maugeri [MM] are in this direction, but their crucial point is different from that of the present paper. They allow the coefficient functions of the principal part A to depend (nonlinearly) also on u, ..., ∇^{m−1}u, on the other hand they request A to satisfy quite a strong ellipticity condition and the lower order term f to fulfill strictly controlled growth conditions, i.e. in (2) \frac{n+4m}{n+2ν} is replaced by \frac{n+4m}{n+2ν} − ε, ε > 0. Then they prove partial regularity of weak solutions. To our knowledge there are no other results concerning the regularity of weak solutions of nonlinear parabolic equations or systems with an elliptic part of arbitrary order. The reason may be that it is difficult to find appropriate testing vectors if the elliptic part is in general form.

Here we employ a different method which is explained in what follows. We show that, in the semilinear case, it is possible to omit the sign condition (3) and to treat systems too. More precisely: A is assumed to be a uniformly elliptic matrix differential operator of order 2m in divergence form, elliptic in the sense of Legendre-Hadamard. For simplicity A is supposed to depend not on t. On f only the controllable growth condition (2) is imposed. Then any weak solution of the initial-boundary value problem for the system (1) is smooth.

The central idea of our proof is to reconstruct u as a strong solution. The main problem we have to overcome is the lack of uniqueness for the system (1). Consequently, instead of (1), we consider a modified system, which has the unique weak solution u and which, on the other hand, allows the construction of a global strong solution. The last step is carried out by using a continuity method with time t as parameter of continuity. Doing so we make extensive use of the properties of the given weak solution u. The fundamental tool from linear theory is a maximal L^{r}-regularity result ([CaV], [Si], see Lemma 4) for abstract evolution equations in Hilbert spaces.

The choice of the modified problem is quite comparable to that taken in [Wa1] in the case of a single equation. In [Wa1], however, the parameter of continuity was selected in a different way. Therefore the results in [Wa1] are essentially restricted to a single equation under the sign condition (3). It is the combination of two particular choices which makes a proof possible in the present case: The choice of the modified problem and the choice of time as parameter of continuity. The latter can be found in [Wa2, chapt. V] already. But to our knowledge this particular combination has not been used previously to prove regularity results.

In section 2 we give the precise formulation of our assumptions and the definition of weak solutions and we present the regularity result. Section 3 is devoted to its proof as sketched above.

2. The Result

Most of our notation is standard; ∥.∥_{k,p} denotes the norm in the vector-valued Sobolev space H^{k,p}(Ω) := H^{k,p}(Ω, IR^{N}) and (., .) the duality product between L^{p}(Ω, IR^{N}) and L^{p′}(Ω, IR^{N}).

We will prove the regularity result under the following assumptions:

(A I) n, m, N ∈ IN, n ≥ 3. Ω ⊂ IR^{n} is a bounded smooth domain of class C^{4m} with outer unit normal ν. For simplicity we assume n > 2m.

(A II) A = ∑_{|α|,|β|≤m}(−1)^{|α|}D^{α}(A_{αβ}(x)D^{β}) is a positive uniformly elliptic

matrix differential operator, i.e.: A_{αβ} : Ω̄ → IR^{N×N} are matrices of class C^{m}(Ω̄), where α, β ∈ IN_{0}^{n} are multiindices of length n, D^{α} = ∏_{j=1}^{n}(\frac{∂}{∂x_{j}})^{α_{j}}.

There is a constant M > 0 such that

M|ξ|^{2m}|ζ|^{2} ≥ ∑_{i,j=1}^{N} ∑_{|α|=|β|=m} A_{αβ}^{ij}(x)ξ^{α}ξ^{β}ζ_{i}ζ_{j} ≥ M^{−1}|ξ|^{2m}|ζ|^{2}

holds for all x ∈ Ω̄, ξ ∈ IR^{n}, ζ ∈ IR^{N}.

Without loss of generality we may assume Gårding’s inequality

∑_{|α|,|β|≤m}(A_{αβ}D^{β}u, D^{α}u) ≥ C_{0}∥u∥_{m,2}^{2} for all IR^{N}-vector functions u ∈ H_{0}^{m,2}(Ω)

with a positive constant C_{0}.

(A III) Let k_{j} be the number of multiindices α with |α| = α_{1} + ... + α_{n} = j.

f : IR_{0}^{+} × Ω̄ × IR^{N} × IR^{N·k_{1}} × ... × IR^{N·k_{m}} ⟶ IR^{N}

is a continuous function, satisfying the growth condition

|f(t, x, p_{0}, ..., p_{m})| ≤ K(1 + ∑_{ν=0}^{m} |p_{ν}|^{\frac{n+4m}{n+2ν}}).

In order to be able to estimate the nonlinear terms we prove the following lemma.

Lemma 1. Let w_{1}, w_{2}, w_{3} ∈ H_{0}^{m,2}(Ω). For ν = 0, ..., m let γ_{ν} be real numbers satisfying the conditions:

\frac{ν(8m − 4ν)}{n + 2ν} ≤ γ_{ν} ≤ \frac{m(8m − 4ν)}{n + 2ν} and 0 < γ_{ν} < 2m.

Suppose that there is constant L such that ∥w_{1}∥_{0,2}, ∥w_{2}∥_{0,2} ≤ L. Then we have

∑_{ν=0}^{m} ∫_{Ω} ||∇^{ν}w_{1}|^{\frac{n+4m}{n+2ν}} − |∇^{ν}w_{2}|^{\frac{n+4m}{n+2ν}} | · |w_{3}| dx

≤ C ∑_{ν=0}^{m} (∥w_{1}∥m,2^{\frac{γν}{2m}} + ∥w_{2}∥m,2^{\frac{γν}{2m}}) ∥w_{1} − w_{2}∥_{m,2} · ∥w_{3}∥_{m,2}^{1− \frac{γν}{2m}} · ∥w_{3}∥0,2^{\frac{γν}{2m}} ,

where C only depends on N, n, m, Ω, L and γ_{ν} (ν = 0, ..., m).

Proof. We remark that there exist numbers γ_{ν} as above, e.g. we may put γ_{0} = \frac{2m^{2}}{n} , γ_{ν} = \frac{ν(8m−4ν)}{n+2ν} for ν = 1, ..., m. By virtue of n > 2m we have 0 < γ_{ν} < 2m.

For a, b ≥ 0 the mean value theorem gives

|a^{\frac{n+4m}{n+2ν}} − b^{\frac{n+4m}{n+2ν}} | ≤ \frac{n + 4m}{n + 2ν} (a^{\frac{4m−2ν}{n+2ν}} + b^{\frac{4m−2ν}{n+2ν}} ) |a − b|.

From this there arises

∑_{ν=0}^{m} ∫_{Ω} ||∇^{ν}w_{1}|^{\frac{n+4m}{n+2ν}} − |∇^{ν}w_{2}|^{\frac{n+4m}{n+2ν}} | · |w_{3}| dx

≤ C(m, n) ∑_{ν=0}^{m} ∫_{Ω} (|∇^{ν}w_{1}|^{\frac{4m−2ν}{n+2ν}} + |∇^{ν}w_{2}|^{\frac{4m−2ν}{n+2ν}} ) · |∇^{ν}(w_{1} − w_{2})| · |w_{3}| dx

≤ C(m, n) ∑_{ν=0}^{m} (∥∇^{ν}w_{1}∥_{0, \frac{4m−2ν}{n+2ν} · \frac{2n}{4m−γν −2ν}} + ∥∇^{ν}w_{2}∥_{0, \frac{4m−2ν}{n+2ν} · \frac{2n}{4m−γν −2ν}} )^{\frac{4m−2ν}{n+2ν}}

· ∥∇^{ν}(w_{1} − w_{2})∥_{0, \frac{2n}{n−2m+2ν}} · ∥w_{3}∥_{0, \frac{2n}{n−2m+γν}} ,

where we applied Hölder’s inequality with exponents p_{ν,1} = \frac{2n}{4m−γ_{ν}−2ν} ≥ \frac{2n}{4m} > 1, p_{ν,2} = \frac{2n}{n−2m+2ν} ≥ 2, p_{ν,3} = \frac{2n}{n−2m+γ_{ν}} > 2; \frac{1}{p_{ν,1}} + \frac{1}{p_{ν,2}} + \frac{1}{p_{ν,3}} = 1. For ν = m we have \frac{4m−2ν}{n+2ν} · p_{ν,1} = p_{ν,2} = 2, 2 ≤ p_{ν,3} < \frac{2n}{n−2m}; i.e. ∇^{m}w_{i} only occurs in the L^{2}-norm, these terms don’t need to be interpolated. For 0 ≤ ν < m we have 2 ≤ \frac{4m−2ν}{n+2ν} · p_{ν,1} ≤ \frac{2n}{n−2(m−ν)} = p_{ν,2}, 2 < p_{ν,3} < \frac{2n}{n−2m}; p_{ν,2} is the limiting exponent for the imbedding H^{m,2}(Ω) ↪ H^{ν,p}(Ω).

Now we employ the general imbedding and interpolation inequality ([Fr, p. 27]):

∥∇^{ν}u∥_{p} ≤ C(∥u∥_{m,2})^{a}(∥u∥_{0,2})^{1−a}.

Here ν is an integer, 0 ≤ ν < m, a ∈ [\frac{ν}{m}, 1] is a real number, m − \frac{n}{2} − ν is not a nonnegative integer and \frac{1}{p} = \frac{n+2ν}{2n} − a\frac{m}{n}. The constant C only depends on m, n, N, Ω, a, ν.

We remark that the restrictions on γ_{ν} ensure all exponents a to be admissible. We obtain:

∑_{ν=0}^{m} ∫_{Ω} ||∇^{ν}w_{1}|^{\frac{n+4m}{n+2ν}} − |∇^{ν}w_{2}|^{\frac{n+4m}{n+2ν}} | · |w_{3}| dx

≤ ∑_{ν=0}^{m} C(m, n, N, Ω, ν, γ_{ν}) (∥w_{1}∥_{m,2}^{\frac{n+2ν}{4m−2ν} · \frac{γν}{2m}} ∥w_{1}∥_{0,2}^{1− \frac{n+2ν}{4m−2ν} · \frac{γν}{2m}}

+∥w_{2}∥_{m,2}^{\frac{n+2ν}{4m−2ν} · \frac{γν}{2m}} · ∥w_{2}∥_{0,2}^{1− \frac{n+2ν}{4m−2ν} · \frac{γν}{2m}} )^{\frac{4m−2ν}{n+2ν}}

· ∥w_{1} − w_{2}∥_{m,2} · ∥w_{3}∥_{m,2}^{1− \frac{γν}{2m}} · ∥w_{3}∥0,2^{\frac{γν}{2m}}

≤ ∑_{ν=0}^{m} C(m, n, N, Ω, L, ν, γ_{ν})(∥w_{1}∥m,2^{\frac{γν}{2m}} + ∥w_{2}∥m,2^{\frac{γν}{2m}})∥w_{1} − w_{2}∥_{m,2}

· ∥w_{3}∥_{m,2}^{1− \frac{γν}{2m}} ∥w_{3}∥0,2^{\frac{γν}{2m}} .

Definition. Let ϕ ∈ L^{2}(Ω), u : (0, T) × Ω → IR^{N} be of class L^{2}((0, T), H_{0}^{m,2}(Ω)) ∩ L^{∞}((0, T), L^{2}(Ω)). u is called a weak solution of the initial boundary value problem

\frac{∂}{∂t}u(t, x) + Au(t, x) + f(t, x, u, ..., ∇^{m}u) = 0, (t, x) ∈ (0, T) × Ω,

(\frac{∂u}{∂ν})^{j}|_{∂Ω} = 0, j = 0, ..., m − 1, t ∈ [0, T], (4)

u(0, x) = ϕ(x), x ∈ Ω,

if the relation

− ∫_{0}^{T} (u(s), χ′(s)) ds + ∑_{|α|,|β|≤m} ∫_{0}^{T}(A_{αβ}D^{β}u(s), D^{α}χ(s)) ds

+ ∫_{0}^{T}(f(s, ., u, ..., ∇^{m}u), χ(s)) ds = (ϕ, χ(0))

holds for all IR^{N}-valued functions χ ∈ L^{2}((0, T), H_{0}^{m,2}(Ω)) with χ′ ∈ L^{2}((0, T), L^{2}(Ω)), χ(T) = 0.

Remark. By Sobolev’s imbedding theorem in one dimension, we have χ ∈ C^{0}([0, T], L^{2}(Ω)), so χ(0), χ(T) make sense. Furthermore

∫_{0}^{T}(f(s, ., u, ..., ∇^{m}u), χ(s)) ds is well defined, this is verified using (A III) and Lemma 1. Namely, we put w_{1} := u, w_{2} := 0, w_{3} := χ and obtain

| ∫_{0}^{T}(f(s, ., u, ..., ∇^{m}u), χ(s)) ds|

≤ C(ess sup_{0<s<T} ∥u(s)∥_{0,2})

· ∫_{0}^{T} \{∥χ(s)∥_{0,2} + ∑_{ν=0}^{m} ∥u(s)∥_{m,2}^{1+ \frac{γν}{2m}} · ∥χ(s)∥_{m,2}^{1− \frac{γν}{2m}} · ∥χ(s)∥0,2^{\frac{γν}{2m}} \} ds

≤ C( ess sup_{0<s<T} ∥u(s)∥_{0,2}) · (1 + ∑_{ν=0}^{m}(∫_{0}^{T} ∥u(s)∥_{m,2}^{2} ds)^{\frac{1}{2} + \frac{γν}{4m}})

·\{(∫_{0}^{T} ∥χ(s)∥_{m,2}^{2} ds)^{\frac{1}{2}} + ess sup_{0<s<T} ∥χ(s)∥\},

the numbers γ_{ν} ∈ (0, 2m), ν = 0, ..., m are defined in Lemma 1. This shows further, that t ↦ f(t, ., u, ..., ∇^{m}u) is in [L^{∞}((0, T), L^{2}(Ω))∩L^{2}((0, T), H_{0}^{m,2}(Ω))]′.

Now we are able to state our main result:

Theorem. Let the assumptions (A I)-(A III) be satisfied, ϕ ∈ H^{2m,2}(Ω)∩ H_{0}^{m,2}(Ω), u ∈ L^{2}((0, T), H_{0}^{m,2}(Ω)) ∩ L^{∞}((0, T), L^{2}(Ω)) be a weak solution of the initial boundary value problem (4).

Then for every 1 < r < ∞ we have

u ∈ L^{r}((0, T), H^{2m,2}(Ω) ∩ H_{0}^{m,2}(Ω)),
u′ ∈ L^{r}((0, T), L^{2}(Ω)).

Remark. Using the theory of L^{p}-semigroups, one can obtain higher regularity, namely u ∈ L^{r}((0, T), H^{2m,p}(Ω)) and under appropriate smoothness conditions on A, f and ϕ also u ∈ C^{μ}([0, T], C^{2m,μ}(Ω̄)), u′ ∈ C^{μ}([0, T], C^{μ}(Ω̄)), cf. the corresponding remark in [Wa1].

But in this paper we want to confine ourselves to L^{2}-theory because the linear results needed here can be proved in a quite elementary manner. The constants in the a-priori estimates as in Lemma 4 below can be calculated explicitly, in particular their dependence on the time interval [0, T]. They can’t blow up as T → 0, this fact is important for the proof below, see section 3.5.

3. Proof of the theorem

3.1. For T̂ > 0 let us consider the following modified initial boundary value problem:

\frac{∂}{∂t}w(t, x) + Aw(t, x) + F (t, x, w, ..., ∇^{m}w) = 0, (t, x) ∈ (0, T̂) × Ω,

(\frac{∂}{∂ν})^{j}w(t)|_{∂Ω} = 0, j = 0, ..., m − 1, (5)

w(0) = ϕ,

where F (t, x, p_{0}, ..., p_{m}) = K · (1 + ∑_{ν=0}^{m} |p_{ν}|^{\frac{n+4m}{n+2ν}}) · q(t, x), q : [0, T̂] × Ω̄ → IR^{N},

q(t, x) = \{ \begin{matrix}\frac{1}{K(1 + ∑_{ν=0}^{m} |∇^{ν}u(t, x)|^{\frac{n+4m}{n+2ν}})}f(t, x, u, ..., ∇^{m}u), \\ if t ∈ [0, T] ∩ [0, T̂], \\ 0, if t ∈ (T, ∞) ∩ [0, T̂]. \end{matrix}

We have |q(t, x)| ≤ 1, so that a weak solution of (5) is defined in exactly the same way as for (4) above.

Interpreting F as inhomogeneity, linear theory (see [Li, pp. 52-55]) yields: w ∈ C^{0}([0, T̂], L^{2}(Ω)), where w is any weak solution of (5). To apply [Li], a standard approximation procedure (truncation of F in each summand) is needed. The truncated nonlinearities with the given weak solution w serve as inhomogeneous terms. Then we take the energy equality for the difference of two approximants. Application of Lemma 1 to the truncated nonlinearities in particular shows the convergence of the approximants in C^{0}([0, T̂], L^{2}(Ω)). Note now, that linear systems like w̃′ + Aw̃ + F = 0 have at most one weak solution, because A is positive. Moreover, the energy equality holds, i.e. if w_{1}, w_{2} ∈ L^{2}((0, T̂), H_{0}^{m,2}(Ω)) ∩ L^{∞}((0, T̂), L^{2}(Ω)) are weak solutions of (5), then for t ∈ [0, T̂] we have

\frac{1}{2}∥(w_{1} − w_{2})(t)∥_{0,2}^{2} + ∫_{0}^{t} ∑_{|α|,|β|≤m}(A_{αβ}D^{β}(w_{1} − w_{2}), D^{α}(w_{1} − w_{2}))(s) ds

(6)

= − ∫_{0}^{t}(F (s, ., w_{1}, ..., ∇^{m}w_{1}) − F (s, ., w_{2}, ..., ∇^{m}w_{2}), w_{1}(s) − w_{2}(s)) ds.

3.2. Lemma 2. For given ε > 0 we have

|(F (t, ., w_{1}, ..., ∇^{m}w_{1}) − F (t, ., w_{2}, ..., ∇^{m}w_{2}), w_{1}(t) − w_{2}(t))|

≤ C(n, m, N, Ω, K, ∥w_{1}∥_{0,2}, ∥w_{2}∥_{0,2}, ε)(∥w_{1}∥_{m,2}^{2} + ∥w_{2}∥_{m,2}^{2})∥w_{1} − w_{2}∥_{0,2}^{2}

+ε∥w_{1} − w_{2}∥_{m,2}^{2}.

Proof. We use Lemma 1 with w_{3} := w_{1}(t) − w_{2}(t); the numbers γ_{ν} ∈ (0, 2m) are defined there. Because |q(t, x)| ≤ 1 we have

|(F (t, ., w_{1}, ..., ∇^{m}w_{1}) − F (t, ., w_{2}, ..., ∇^{m}), w_{1}(t) − w_{2}(t))|

≤ K · ∑_{ν=0}^{m} ∫_{Ω} | |∇^{ν}w_{1}(t)|^{\frac{n+4m}{n+2ν}} − |∇^{ν}w_{2}(t)|^{\frac{n+4m}{n+2ν}} | · |w_{1}(t) − w_{2}(t)| dt

≤ C(∥w_{1}∥_{0,2}, ∥w_{2}∥_{0,2})\{∑_{ν=0}^{m}(∥w_{1}∥m,2^{\frac{γν}{2m}} + ∥w_{2}∥m,2^{\frac{γν}{2m}}) · ∥w_{1} − w_{2}∥_{m,2}^{2− \frac{γν}{2m}}

·∥w_{1} − w_{2}∥0,2^{\frac{γν}{2m}} \}

≤ ε∥w_{1} − w_{2}∥_{m,2}^{2} + C(ε, ∥w_{1}∥_{0,2}, ∥w_{2}∥_{0,2}) · (∥w_{1}∥_{m,2}^{2} + ∥w_{2}∥_{m,2}^{2}) ·∥w_{1} − w_{2}∥_{0,2}^{2}.

An immediate consequence of the energy equality (6), Lemma 2, Gårding’s inequality and Gronwall’s lemma is the following uniqueness result.

Lemma 3. The initial boundary value problem (5) has at most one weak solution w ∈ L^{2}((0, T̂), H_{0}^{m,2}(Ω)) ∩ L^{∞}((0, T̂), L^{2}(Ω)).

Remark. u is a weak solution of (5) on [0, T] and hence coincides with the unique weak solution of (5) on [0, T]. Furthermore we have u ∈ C^{0}([0, T], L^{2}(Ω)).

3.3. In what follows we repeatedly have to interpolate ∥∇^{ν}v∥_{0,p} between ∥v∥_{0,2} and ∥v∥_{2m,2} according to the general interpolation and imbedding inequality [Fr, p. 27]. We remark that this inequality also holds in the case 2m − \frac{n}{2} − ν ∈ IN_{0}, if p is not too large, e.g. if p ≤ \frac{2n}{n−2m}.

Our aim is to construct a strong solution to (5) on an interval larger than [0, T]. So for the following we fix T̂ > T, e.g. we may put T̂ := T + 1.

We consider A as closed operator in L^{2}(Ω) with domain D(A) = H^{2m,2}(Ω)∩ H_{0}^{m,2}(Ω). It is well known that −A generates an analytic semigroup in L^{2}(Ω), which exponentially decays. For the rest of the proof we fix without loss of generality r > \frac{n+4m}{2m}. In particular we have r > 2. The crucial tool from linear theory is the following result.

Lemma 4. Let 0 < T̃ ≤ T̂, f ∈ L^{r}((0, T̃), L^{2}(Ω)), 1 − \frac{1}{r} < κ ≤ 1, Ψ ∈ D(A^{κ}). Then there is exactly one solution v ∈ L^{r}((0, T̃), H^{2m,2}(Ω) ∩ H_{0}^{m,2}(Ω)), v′ ∈ L^{r}((0, T̃), L^{2}(Ω)) of

v′ + Av = f, v(0) = Ψ.

Moreover the following a-priori-estimate holds:

∫_{0}^{T̃} ∥v(s)∥_{2m,2}^{r} ds + ∫_{0}^{T̃} ∥v′(s)∥_{0,2}^{r} ds

≤ C_{1}{∥A^{κ}Ψ∥_{0,2}^{r} + ∫_{0}^{T̃} ∥f(s)∥_{0,2}^{r} ds},

where C_{1} is a constant, which only depends on r, Ω, A, T̂.

This was shown first within an abstract framework in [Si] if Ψ = 0. It’s easily checked that Ψ ∈ D(A^{κ}) with 1 − \frac{1}{r} < κ ≤ 1 is sufficient for the estimate in question. This condition may be relaxed, cf. [CaV]. Lemma 4 is valid for any r ∈ (1, ∞), because L^{2}(Ω) is a Hilbert space. For the estimate in question with L^{2}(Ω) replaced by L^{p}(Ω) see [Wa4], [CaV], [Wa3, chapt. I.10 and II.3].

3.4. The following estimate for the nonlinearity F will be used for proving the existence of a local strong solution of (5).

Lemma 5. Let v, w ∈ L^{r}((0, T̃), H^{2m,2}(Ω)), v′, w′ ∈ L^{r}((0, T̃), L^{2}(Ω)). For every ε > 0 we have

∫_{0}^{T̃} ∥F (s, ., v, ..., ∇^{m}v) − F (s, ., w, ..., ∇^{m}w)∥_{0,2}^{r} ds

≤ C(r){ε(∑_{ν=0}^{m} sup_{s∈[0,T̃]} (∥v(s)∥_{0,2}^{α_{ν}} + ∥w(s)∥_{0,2}^{α_{ν}}))

· ∫_{0}^{T̃} ∥v(s) − w(s)∥_{2m,2}^{r} ds}

+C(ε, n, m, N, Ω, K, r)( sup_{s∈[0,T̃]} ∥v(s) − w(s)∥_{0,2}^{r})

· ∫_{0}^{T̃}(∥v(s)∥_{2m,2}^{r} + ∥w(s)∥_{2m,2}^{r}) ds

with some positive numbers α_{ν} > 0.

Proof. We use similar arguments as in the proof of Lemma 1. For ν = 0, ..., m we fix real numbers p_{ν} ∈ (2, ∞) satisfying

\frac{2n + 4ν}{n + 4ν − 4m} ≥ p_{ν} ≥ \frac{2n^{2} + 4nν}{n^{2} + 4nν + 4ν^{2} − 4mn + 16m^{2} − 16mν} ,

if ν > 2m − \frac{n}{2},

min\{\frac{2n + 4ν}{n + 4ν − 4m}, \frac{2n}{n − 2m}\} ≥ p_{ν} ≥ \frac{2n^{2} + 4nν}{n^{2} + 4nν + 8m^{2} − 4mn − 4mν} ,

if ν > m − \frac{n}{4} and ν ≤ 2m − \frac{n}{2},

\frac{2n}{n − 2m} ≥ p_{ν} ≥ \frac{2n^{2} + 4nν}{n^{2} + 4nν + 8m^{2} − 4mn − 4mν} , if ν ≤ m − \frac{n}{4}.

We remark that the upper bound for p_{ν} is greater than the lower bound and than 2. These conditions imply

2 ≤ p_{ν} ≤ \frac{2n}{n − 4m + 2ν} ,

2 ≤ \frac{2p_{ν}}{p_{ν} − 2} · \frac{4m − 2ν}{n + 2ν} ≤ \frac{2n}{n − 4m + 2ν} if ν > 2m − \frac{n}{2},

2 ≤ p_{ν} ≤ \frac{2n}{n − 2m},

2 ≤ \frac{2p_{ν}}{p_{ν} − 2} · \frac{4m − 2ν}{n + 2ν} ≤ \frac{2n}{n − 2m} if ν ≤ 2m − \frac{n}{2},

and the interpolation exponents below are admissable. For s ∈ (0, T̃) we have

∥F (s, ., v, ..., ∇^{m}v) − F (s, ., w, ..., ∇^{m}w)∥_{0,2}

≤ K∥ ∑_{ν=0}^{m}(|∇^{ν}v|^{\frac{n+4m}{n+2ν}} − |∇^{ν}w|^{\frac{n+4m}{n+2ν}})∥_{0,2}

≤ C(m, n, N, K) ∑_{ν=0}^{m} ∥∇^{ν}(v − w)∥_{0,p_{ν}}

·(∥|∇^{ν}v|^{\frac{4m−2ν}{n+2ν}} ∥_{0, \frac{2pν}{pν −2}} + ∥|∇^{ν}w|^{\frac{4m−2ν}{n+2ν}} ∥_{0, \frac{2pν}{pν −2}} )

by Hölder’s inequality

≤ C(m, n, N, K) ∑_{ν=0}^{m} ∥∇^{ν}(v − w)∥_{0,p_{ν}} · (∥∇^{ν}v∥0, \frac{2pν}{pν −2} · \frac{4m−2ν}{n+2ν}^{\frac{4m−2ν}{n+2ν}}

+∥∇^{ν}w∥0, \frac{2pν}{pν −2} · \frac{4m−2ν}{n+2ν}^{\frac{4m−2ν}{n+2ν}})

≤ ∑_{ν=0}^{m} C(m, n, N, K, Ω, ν, p_{ν})(∥v∥_{0,2}^{\frac{4m−2ν}{n+2ν} (1−a_{ν})}∥v∥_{2m,2}^{\frac{4m−2ν}{n+2ν} a_{ν}}

+∥w∥_{0,2}^{\frac{4m−2ν}{n+2ν} (1−a_{ν})}∥w∥_{2m,2}^{\frac{4m−2ν}{n+2ν} a_{ν}}) · ∥v − w∥_{0,2}^{(1− \frac{ν}{2m} − \frac{n}{4m} + \frac{n}{2mpν} )}
·∥v − w∥_{2m,2}^{( \frac{ν}{2m} + \frac{n}{4m} − \frac{n}{2mpν} )},

where a_{ν} = \frac{ν}{2m} + \frac{n}{4m} − \frac{n}{2m} · \frac{p_{ν}−2}{2p_{ν}} · \frac{n+2ν}{4m−2ν} and where we used the general interpolation and imbedding inequality [Fr, p. 27]; we observe that a_{ν} \frac{4m−2ν}{n+2ν} = 1 − \frac{ν}{2m} − \frac{n}{4m} + \frac{n}{2mp_{ν}}. Hence

∥F (s, ., v, ..., ∇^{m}v) − F (s, ., w, ..., ∇^{m}w)∥_{0,2}

≤ C(m, n, N, K, Ω, p_{0}, ..., p_{m}) · ∑_{ν=0}^{m}\{(∥v∥_{0,2} + ∥w∥_{0,2})^{( \frac{4m−2ν}{n+2ν})(1−a_{ν})}

· ∥v − w∥_{2m,2}^{( \frac{v}{2m} + \frac{n}{4m} − \frac{n}{2mpν} )}\} · \{(∥v∥_{2m,2} + ∥w∥_{2m,2})

· ∥v − w∥_{0,2}\}^{1− \frac{ν}{2m} − \frac{n}{4m} + \frac{n}{2mpν}}

≤ ε^{\frac{1}{r}} \{∑_{ν=0}^{m}(∥v(s)∥_{0,2}^{β_{ν}} + ∥w(s)∥_{0,2}^{β_{ν}})\} · ∥v(s) − w(s)∥_{2m,2}

+ C(ε, m, n, N, K, Ω, r, p_{0}, ..., p_{m})∥v(s) − w(s)∥_{0,2}(∥v(s)∥_{2m,2} + ∥w(s)∥_{2m,2})

by Young’s inequality; β_{ν} are some positive numbers, determined by n, m, ν, p_{ν}.

We take this estimate to the power r, integrate with respect to s and immediately obtain Lemma 5.

Our local existence result reads as follows.

Lemma 6. Let 1 − \frac{1}{r} < κ ≤ 1, Ψ ∈ D(A^{κ}). Then there exists a positive time T̃ ∈ (0, T̂] and a solution

w ∈ L^{r}((0, T̃), H^{2m,2}(Ω) ∩ H_{0}^{m,2}(Ω)), w′ ∈ L^{r}((0, T̃), L^{2}(Ω))

of the initial boundary value problem (5), where w(0) = Ψ instead of w(0) = ϕ is requested. T̃ depends in particular on ∥A^{κ}Ψ∥_{0,2}, A, Ω, K, r.

Proof. Let ψ(t) ∈ L^{r}((0, T̂), H^{2m,2}(Ω)∩H_{0}^{m,2}(Ω)), ψ′ ∈ L^{r}((0, T̂), L^{2}(Ω)), ψ(0) = Ψ be an arbitrary auxiliary function, e.g. the solution of ψ′+Aψ = 0. We set

M := {w|w ∈ L^{r}((0, T̃), H^{2m,2}(Ω) ∩ H_{0}^{m,2}(Ω)),

w′ ∈ L^{r}((0, T̃), L^{2}(Ω)), w(0) = Ψ,

∫_{0}^{T̃}(∥w(s)∥_{2m,2}^{r} + ∥w′(s)∥_{0,2}^{r}) ds ≤ 1 + C_{2}},

where C_{2} := C_{1}(∥A^{κ}Ψ∥_{0,2}^{r} +2^{r} ∫_{0}^{T̂}∥F (s, ., ψ, ..., ∇^{m}ψ)∥_{0,2}^{r} ds), C_{1} is taken from

Lemma 4, T̃ has to be determined below, without loss of generality we assume T̃ ≤ 1.

The map G : M → M is defined as follows: for w ∈ M, let v = Gw be the solution of

v′(t) + Av(t) = −F (t, ., w, ..., ∇^{m}w),
v(0) = Ψ,
v ∈ L^{r}((0, T̃), H^{2m,2}(Ω) ∩ H_{0}^{m,2}(Ω)),
v′ ∈ L^{r}((0, T̃), L^{2}(Ω)).

First we prove G(M) ⊂ M for sufficiently small T̃. By Lemmata 4 and 5 we have:

∫_{0}^{T̃} (∥v(s)∥_{2m,2}^{r} + ∥v′(s)∥_{0,2}^{r}) ds

≤ C_{1}{∥A^{κ}Ψ∥_{0,2}^{r} + 2^{r} ∫_{0}^{T̃} ∥F (s, ., w, ..., ∇^{m}w) − F (s, ., ψ, ..., ∇^{m}ψ)∥_{0,2}^{r} ds

+2^{r} ∫_{0}^{T̃} ∥F (s, ., ψ, ..., ∇^{m}ψ)∥_{0,2}^{r} ds}

≤ C_{2} + C_{1} · 2^{r} · C(n, m, K, r)\{ε ∑_{ν=0}^{m}((∥Ψ∥_{0,2} + (1 + C_{2})^{\frac{1}{r}})^{α_{ν}}

+ sup_{s∈[0,T]} ∥ψ(s)∥_{0,2}^{α_{ν}}) · (1 + C_{2} + ∫_{0}^{T̂} ∥ψ(s)∥_{2m,2}^{r} ds)}

+C(ε)( sup_{s∈[0,T̃]} ∥ψ(s) − w(s)∥_{0,2}^{r}) · (1 + C_{2} + ∫_{0}^{T̂} ∥ψ(s)∥_{2m,2}^{r} ds).

For an appropriate choice of ε, the second summand is ≤ \frac{1}{2}. Furthermore we have

∥ψ(s) − w(s)∥_{0,2} ≤ ∥ψ(s) − Ψ∥_{0,2} + ∥Ψ − w(s)∥_{0,2}

≤ (∫_{0}^{s} ∥ψ′(s)∥_{0,2}^{r} ds)^{\frac{1}{r}} s^{1− \frac{1}{r}} + (∫_{0}^{s} ∥w′(s)∥_{0,2}^{r} ds)^{\frac{1}{r}} s^{1− \frac{1}{r}}

≤ \{(∫_{0}^{T̂} ∥ψ′(s)∥_{0,2}^{r} ds)^{\frac{1}{r}} + (C_{2} + 1)^{\frac{1}{r}} \}s^{1− \frac{1}{r}} .

Consequently T̃ can be chosen in such a way, that the third summand is also ≤ \frac{1}{2}. Hence v ∈ M.

Now let w_{1} = ψ, w_{n+1} = Gw_{n}. In a completely analogous way it is shown, that

∫_{0}^{T̃}(∥w_{n+2}(s) − w_{n+1}(s)∥_{2m,2}^{r} + ∥w′_{n+2}(s) − w′_{n+1}(s)∥_{0,2}^{r}) ds

≤ ϱ · ∫_{0}^{T̃}(∥w_{n+1}(s) − w_{n}(s)∥_{2m,2}^{r} + ∥w′_{n+1}(s) − w′_{n}(s)∥_{0,2}^{r}) ds,

with some ϱ < 1 for an appropriate choice of T̃. The contraction mapping principle completes the proof.

3.5. We define T_{max} := sup{T̃| on [0, T̃] there exists a solution w as in Lemma 6}. In particular there is a unique strong solution w on [0, T_{max}). If we can show T_{max} > T, it will follow that u = w on [0, T], thereby proving the theorem.

We assume the contrary: T_{max} ≤ T. On [0, T_{max}) we have u = w, and u is uniformly continuous on [0, T] in L^{2}(Ω). So there is a δ > 0, which only depends on u, such that for 0 < t < t′ < T_{max}, |t − t′| < δ, |t − t′| < t we have:

C(T̂) · ∑_{ν=0}^{m} ∥w(s) − w(2t − s)∥_{0,2}^{r· \frac{4m−2ν}{n+2ν}} ≤ \frac{1}{2} for all s ∈ [t, t′], (7)

where C(T̂) is the constant below;

ŵ(s) := w(s) − w(2t − s) solves

ŵ′(s) + Aŵ(s) = −F (s, ., w(s), ..., ∇^{m}w(s)) + w′(2t − s) − Aw(2t − s), ŵ(t) = 0;

∫_{t}^{t′}(∥ŵ(s)∥_{2m,2}^{r} + ∥ŵ′(s)∥_{0,2}^{r}) ds

≤ C(T̂){ ∫_{2t−t′}^{t} (∥w′(s)∥_{0,2}^{r} + ∥w(s)∥_{2m,2}^{r}) ds

+ ∫_{t}^{t′}(1 + ∑_{ν=0}^{m} ∥|∇^{ν}w(s) − ∇^{ν}w(2t − s)|^{\frac{n+4m}{n+2ν}} ∥_{0,2}^{r}) ds

+ ∫_{t}^{t′}(∑_{ν=0}^{m} ∥|∇^{ν}w(2t − s)|^{\frac{n+4m}{n+2ν}} ∥_{0,2}^{r}) ds\}

≤ C(T̂)\{ ∫_{2t−t′}^{t} (1 + ∥w′(s)∥_{0,2}^{r} + ∥w(s)∥_{2m,2}^{r}(1 + ∑_{ν=0}^{m} ∥w(s)∥_{0,2}^{r· \frac{4m−2ν}{n+2ν}})) ds

+ ∑_{ν=0}^{m} ∫_{t}^{t′}(∥w(s) − w(2t − s)∥_{2m,2}^{r} · ∥w(s) − w(2t − s)∥_{0,2}^{r· \frac{4m−2ν}{n+2ν}})ds\}

by the general interpolation and imbedding inequality [Fr], p. 27

≤ \frac{1}{2} ∫_{t}^{t′} ∥ŵ(s)∥_{2m,2}^{r} ds + C(T̂) · ∫_{2t−t′}^{t} (1 + ∥w′(s)∥_{0,2}^{r}

+∥w(s)∥_{2m,2}^{r}(1 + ∑_{ν=0}^{m} ∥w(s)∥_{0,2}^{r· \frac{4m−2ν}{n+2ν}})) ds by (7).

Hence

∫_{t}^{t′}(∥w′(s)∥_{0,2}^{r} + ∥w(s)∥_{2m,2}^{r}) ds

≤ C(T̂) ∫_{2t−t′}^{t} (1 + ∥w′(s)∥_{0,2}^{r} + ∥w(s)∥_{2m,s}^{r}(1 + ∑_{ν=0}^{m} ∥w(s)∥_{0,2}^{r· \frac{4m−2ν}{n+2ν}})) ds.

In other words: finiteness of the L^{r} − H^{2m,2}-norm of w and the L^{r} − L^{2}-norm of w′ on [t, t + δ] follows from the finiteness on [t − δ, t]. Starting at t = \frac{1}{2} T_{max} and proceeding with intervals of uniform length δ, we conclude:

w ∈ L^{r}((0, T_{max}), H^{2m,2}(Ω) ∩ H_{0}^{m,2}(Ω))
w′ ∈ L^{r}((0, T_{max}), L^{2}(Ω)).

By Lemma 5 this yields F (., ., w, ..., ∇^{m}w) ∈ L^{r}((0, T_{max}), L^{2}(Ω)). Using the integral equation for w it follows from r > \frac{n+4m}{2m} that

A^{σ}w ∈ C^{0}([0, T_{max}], L^{2}(Ω)) with some σ ∈ (\frac{n + 2m}{n + 4m}, 1),

and by Sobolev we get (1 + ∑_{ν=0}^{m} |∇^{ν}w|^{\frac{n+4m}{n+2ν}}) ∈ C^{0}([0, T_{max}], L^{2}(Ω)). Because

q ∈ L^{∞}((0, T) × Ω̄), we only have F (., ., w, ..., ∇^{m}w) ∈ L^{∞}((0, T_{max}), L^{2}(Ω)).

So in a second step, we obtain A^{κ}w ∈ C^{0}([0, T_{max}], L^{2}(Ω)) where κ > 1−\frac{1}{r}, i.e. w(T_{max}) ∈ D(A^{κ}).

Now applying Lemma 6 to the initial value problem (5) on [T_{max}, T_{max}+T̃], we arrive at the desired contradiction.

Remarks.

1) The proof of the theorem shows, that it is possible to obtain an a-priori bound for any solution u in the space L^{r}((0, T), H^{2m,2}(Ω)), which depends in particular on the modulus of continuity of u with respect to L^{2}(Ω):

ω(r) = sup_{{s,t∈[0,T] \atop |s−t|≤r}} ∥u(t) − u(s)∥_{0,2}.

2) The question is also of interest whether it is possible to obtain a-priori bounds in L^{r}((0, T), H^{2m,2}(Ω)), which only depend on

sup_{s∈[0,T]} ∥u(s)∥_{0,2}^{2} + ∫_{0}^{T} ∥u(s)∥_{m,2}^{2} ds.

We think that an affirmative answer is only possible under strong additional assumptions on f, like

|\frac{∂f_{i}}{∂u_{j}} | ≤ C(1 + |u|^{\frac{4m}{n}}), (8)

if e.g. f only depends on u. Then bounds could be obtained by differentiating the differential equation with respect to t. Condition (8) however is already violated by simple nonlinearities, e.g. in the case N = 2, n > 4m by f(u_{1}, u_{2}) = (u_{2}(1 + |u_{1}|)^{\frac{4m}{n}} , u_{1}(1 + |u_{2}|)^{\frac{4m}{n}}).

At the first glance, the modified nonlinearity F in system (5) seems to be in appropriate form. But we don’t know anything about \frac{∂q}{∂t}; this prevents us from finding a-priori bounds of the type just mentioned.

Acknowledgement. We are grateful to the referees for some helpful and interesting suggestions.

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Hans-Christoph Grunau Wolf von Wahl Lehrstuhl für Angewandte Mathematik Universität Bayreuth D-95440 Bayreuth