An existence theorem for parabolic equations on R N with

We prove existence of solutions for parabolic initial value problems @tu = u + f(u) on R N , where f : R! R is a bounded, but possibly discontinuous function. AMS Classication: 35K57


Introduction
We prove an existence theorem for the following parabolic initial value problem on Q = R N × (0, ∞), where α ∈ BU C(R N ) is a bounded, uniformly continuous function and f : R → R is a bounded, measurable function.Throughout the paper ∆ denotes the Laplacian, ∇ denotes gradient, •, • is the usual inner product in R N and 'measurable' means Borel-measurable.Problems of the above form cover a wide range of models in applied sciences, e.g. in combustion theory and nerve conduction.Our main motivation is the model of best response dynamics arising in game theory [11].In this model f is a differentiable function on [0, 1] \ {a} for some a ∈ (0, 1) and f (0) = f (1) = 0, f (u) < 0, if u ∈ (0, a); f (u) > 0, if u ∈ (a, 1).
(Outside the interval [0, 1] it can be extended as zero.)A typical special case is f (u) = −u + H(u − a), where H is the Heaviside function.Similar problems were investigated by several authors mainly on bounded domains.The equation is usually considered as a differential inclusion.One of the first results in this field was achieved by Rauch [14].He proved the existence of a solution u ∈ L 2 ([0, t * ], H 1 0 (Ω)), where Ω ⊂ R N is a bounded domain and f is locally bounded.In [4] the existence of weak solutions is proved in a similar space.Bothe [2] extended the existence theorem for systems but also considered bounded domains.Terman [17] proved an existence theorem in the one dimensional case for the special nonlinearity f (u) = −u + H(u − a).In his paper the solution is classical at those points (x, t) where u(x, t) = a.In [9] f (u) = g(u) + H(u − 1), g is nonnegative, nondecreasing and locally Lipschitz continuous, and the space domain is [0, π].The existence of a u ∈ C([0, t * ], H 1 0 (0, π)) solution is proved.The problem was studied in more general contexts on bounded domains.In [3] the problem is considered with a nonlinear elliptic operator, in [6] and [15] the case of functional partial differential equations is investigated.The results concerning the case of the whole space R N are mainly for the elliptic case, see e.g.[1,5].For other results concerning existence and uniqueness questions for differential equations with discontinuous nonlinearity we refer to the monographs [10,18] and the references therein.Here we prove that there exists a continuous solution on R N × [0, ∞).We do not restrict ourselves to bounded domains and one space dimension.Moreover, our solutions are not even in L 2 (R N ) (for fixed t), because we would like to treat spatially constant non zero solutions, and travelling waves connecting these, too.Hence none of the methods of the above papers works in itself.We have to combine several ideas to prove the existence theorem.
The usual way of introducing the corresponding differential inclusion is to define the semicontinuous functions We note that if f is continuous in u, then f(u) = f(u) = f (u).Now we can define the notion of a solution.
Definition 1 The function u : (ii) u satisfies the corresponding differential inclusion in the weak sense, that is there exists a bounded measurable function h : The main result of this paper is the following theorem.
Theorem 1 Let α ∈ BU C(R N ) be a bounded, uniformly continuous function and f : R → R be a bounded, measurable function.Then ( 1)-( 2) has a solution in Q.
The proof of the Theorem in Section 3 consists of the following STEPs.
STEP1 Introduction of a sequence (f n ) of C ∞ functions approximating f .STEP2 Solving the approximating equations STEP3 Using the Arzelá-Ascoli theorem we get a uniformly convergent subsequence of (u n ).The solution u is defined as the limit of that subsequence.STEP4 Constructing h as the limit of STEP5 To prove that h satisfies (5).
STEP6 To prove that u is continuously differentiable w.r.t.x.

Preliminaries
We will consider our problem as an abstract evolution equation.Let X = BU C(R N ) be the space of bounded, uniformly continuous functions endowed with the supremum norm, • .For ψ ∈ X we define where We will use the following properties of {T (t)} t≥0 .
5. There exists a > 0, such that for any 0 < τ 1 < τ 2 there holds T (τ Proof.The first four statements are well-known, see e.g.[8,12], the last one will be proved. Let us denote the generator of the analytic semigroup T (t) by A. Then by the analyticity (see [7] Ch.II.Theorem 4.6) there exists a > 0 such that Let ψ ∈ X, ψ = 1.Using the above formula and (1.7) in [7] Ch.II.we obtain The last step follows from the simple inequality It is also well-known that the solution of the inhomogeneous problem can be expressed in the abstract framework.Let us assume that h : R N × [0, t * ]) → R is bounded and uniformly continuous for some t * > 0, and let us introduce H : [0, t * ] → X, H(t)(x) = h(x, t).Then the solution of the inhomogeneous problem ( 8)-( 9) takes the form v(x, t) = V (t)(x), where The required regularity of V is proved in the next two Propositions.These statements can be proved in the abstract setting (see e.g.[13]), but here we prove them in our special case to make the paper self-contained.The uniform continuity of V follows from 3. of Proposition 1 and from the following statement.
Proposition 2 Let H : [0, t * ] → X be continuous and where EJQTDE, 2001 No. 8, p. 3 Proof.Let us assume t 1 < t 2 .Then For the norm of the second term we have To estimate the norm of the first term we use 5. of Proposition Using ( 13), ( 14) and ( 15) we obtain the desired inequality. 2 We will need that the boundedness of the function h implies the differentiability of v w.r.t.x.The notation ∂ k will be used for the partial derivative w.r.t.x k .
Proposition 3 Let α ∈ BU C(R N ) and h be bounded and Borel-measurable in Q.Then the function is continuously differentiable w.r.t.x, and for its partial derivatives we have Moreover, for any 0 < t 1 < t 2 the partial derivatives are bounded in R N × [t 1 , t 2 ]: Proof.The first statement follows from the theorem on differentiation of parametric integrals.Since α and h are bounded, (17) follows easily from the formulas below: which can be verified by direct integration, using that 2 Now we summarize the equicontinuity results concerning the solution of the inhomogeneous equation.

EJQTDE, 2001
No. 8, p. 4 Proposition 4 Let α ∈ BU C(R N ), h be bounded and measurable in Q and v be defined in (16).Then for all ε > 0 and t * > 0 there exists δ > 0 (depending only on α and h ), such that for all t 1 , t 2 ∈ [0, t * ] and Proof.We will prove that for all ε > 0 and t * > 0 there exists δ > 0, such that and Inequality (18) follows from Proposition 1 and Proposition 2. Namely, with H(t)(x) = h(x, t), V (t)(x) = v(x, t) and using (10) and ( 12) one obtains sup According to 3. of Proposition 1 for any ε > 0 there exists δ 1 > 0 (depending only on α) such that for According to Proposition 2 there exists δ 2 > 0 (depending only on h and t * ) such that for Thus (18) follows from (20)-( 22) with δ = min{δ 1 , δ 2 }.Now let us turn to the verification of (19).Let us introduce Then by 4. of Proposition 1 there exists δ 1 > 0 (depending only on α) such that According to Proposition 3 Hence there exists δ 2 > 0 (depending only on h and t * ) such that It is easy to prove that the boundedness and local Lipschitz continuity of g implies the same properties for G.The following existence theorem is proved in [16] ( Theorem 11.12.).
Proposition 5 Let α ∈ X and G be bounded and locally Lipschitz continuous.Then there exists a continuous solution U : [0, ∞) → X of (27).