Generalized Solutions of Hamilton – Jacobi Equation to a Molecular Genetic Model

. A boundary value problem with state constraints is under consideration for a nonlinear noncoercive Hamilton-Jacobi equation. The problem arises in molecular biology for the Crow – Kimura model of ge-netic evolution. A new notion of continuous generalized solution to the problem is suggested. Connections with viscosity and minimax generalized solutions are discussed. In this paper the problem is studied for the case of additional requirements to structure of solutions. Constructions of the solutions with prescribed properties are provided and justiﬁed via dynamic programming and calculus of variations. Results of simulations are exposed.


Introduction
In [1] a new way to study molecular evolution has been proposed.According to this way dynamics of the Crow -Kimura model for molecular evolution can be analyzed via the following HJE ∂u/∂t + H(x, ∂u/∂x) = 0, (1) where the Hamiltonian H(•) has the form The function f (•) in ( 2) is given and called fitness.Equation ( 1) is considered for t ≥ 0, −1 ≤ x ≤ 1.It is also assumed that an initial function u 0 : R → R is given such that u(0, x) In [1] problem (1)-( 3) was studied for input data u 0 (x) = −a(x − x 0 ) 2 , a > 0, f (x) = x 2 and physical interpretations were used.
The classical method for solving PDE of the first order in Cauchy problem is the method of characteristics (see, e.g.[2]).This method reduces integration of PDEs to integration of the characteristic system of ODEs.
The characteristic system for problem Here H x (x, p) = ∂H(x, p)/∂x, H p (x, p) = ∂H(x, p)/∂p, f (x) = ∂f (x)/∂x.Solutions of the system (4)-( 5) are called characteristics.Components x(•, y), p(•, y) and z(•, y) of the solution are called state, conjugate, and value characteristics, respectively.The method of characteristics can be applied to constructions of solutions for problem (1)-( 3) in such a neighborhood of the initial manifold (5) where state characteristics don't cross.As a rule, characteristics for problem (1)-( 3) are nonextendable to the whole time axis and can cross each other.Moreover, there are points in strip t ≥ 0, −1 ≤ x ≤ 1. where solution of ( 1)-(3) should be found, and where the state characteristics do not pass.An example of such a behavior of state characteristics is presented on Fig. 1.
So, one can see that solutions of the problem (1)-( 3) should be understood in a generalized sense.
In [3], we introduced a concept of continuous generalized solutions (see Definition 1 below) and proved it's existence in problem (1)-(3) using tools of Nonsmooth Analysis and results of the Optimal Control Theory.It was also shown that the generalized solution is not unique.
In this paper, we consider problem (1)-( 3) with additional requirements to the structure of solutions, see [4] and [5].Namely, we need to construct a continuous solution in the strip t ≥ 0, −1 ≤ x ≤ 1 in such a way that it coincides with a solution obtained by the method of characteristics in a domain part where the characteristics defined by ( 4) and ( 5) pass.
The paper is organized as follows.In Section 2, the definition of a continuous generalized solution is introduced, and the results on its existence are presented.In Section 3 we state the problem of constructing the generalized solution with prescribed properties, give sufficient conditions under which the problem can be solved, and formulate auxiliary results on which solving is based.A scheme for constructing the generalized solution and results of a simulation are presented in Section 4 and Section 5 respectively.And, in Section 6, we compare our generalized solution with viscosity solutions.Let T > 0 be such an instant that characteristics (4), (5) are extendable up to T , and x(•, y), p(•, y), z(•, y) are continuous on [0, T ] for all y ∈ [−1; 1].Exact estimates for intervals of extendibility are obtained in [4,6].
We consider problem (1)-(3) on the restricted closed domain and also use the notations In the HJEs' theory various concepts of generalized solutions have been introduced (see, e.g.[7][8][9]).Note that definitions of generalized solutions to HJEs in open areas were applied to problems with state constraints as additional requirements to solutions on the border were imposed.These requirements play a role of boundary conditions.Unfortunately, results of the theories of generalized solutions are inapplicable to the problem (1)- (3).In particular, one of the key conditions under which the known theorems on existense of a generalized viscosity solution [8,10] has been proved is the coercivity of the Hamiltonian (see (22) below).And the theory of minimax solutions [9] is not developed for problems with state constraints.So, below a new definition of a generalized solution is introduced [3].This definition is based on the minimax and viscosity approahes and uses the following tools of nonsmooth analysis [10,11].
Let W be a set in R 2 .Denote by W the closure of this set, by C(W ) -the class of functions continuous on the set W .
Let u(•) ∈ C(W ) and (t, x) ∈ W .The subdifferential of the function u(•) at (t, x) is the set .
The superdifferential of the function u(•) at (t, x) is the set Let Dif(u) be the set of points where the function u(•) ∈ C(W ) is differentiable.For a given set M ⊂ R 2 , the symbol coM means its convex hull [13] 3) iff it satisfies the initial condition (3) and the following relations are true

Existence of Generalized Solutions
The following statement was proved in [4] by using tools of Mathematical Theory of Optimal Control [14] and the method of generalized characteristics [15,16].
To obtain u(t, x) in accordance with (10), one should consider the set of all state characteristics x(•, y ) passing through the point (t, x), namely, x(t, y ) = x.Note that the generalized solution to problem (1)-( 3) is not unique because of wide choice of functions ϕ(•) in Theorem 1.

Solutions with Prescribed Properties
Here, we consider a problem to construct the generalized solution of some particular structure.
Let x − (t) = x(t, −1) and x + (t) = x(t, +1), t ∈ [0, T ] be the state characteristics started at t = 0 from the points x = −1 and x = 1, respectively.Below, we assume that the following condition is satisfied.
A. For the state characteristics x(•, y) with initial conditions (5) at t = 0 the inequalities are valid

Define the subdomains
So, under the assumption A, we get The goal of the work is to construct the generalized solution to problem (1)-( 3) such that it has the following form in G 0 : where x(t) = x(t, y), p(t) = p(t, y), t ≥ 0, are state and conjugate characteristics, respectively, which satisfy at t = 0 the initial conditions x(0, y) = y, p(0, y) = u 0 (y), y ∈ [−1, 1].

Auxiliary Problems of Calculus of Variations
Consider the following two problems of Calculus of Variations over the set of all continuously differentiable functions x(•) where The following assertions are proven in [4,5], where conditions B1-B2 are essential.

Numerical example
Results of simulation for the input data u 0 (x) = −0.02x 2 + 0.001 cos 2πx, f (•) = −0.5x 2 are presented in Fig. 2. One can easily check that these input data satisfy the conditions B1, B2.

Comparison with Viscosity Solution
One can see that Definition 1 coincides with the definition of viscosity solution in the interior points of the region Π T .The difference between these definitions is evident at boundary points, namely, on the set Γ T .In condition (8), the inequality holds for such points (a, s) of the subdifferential D − u(t, x) which at the same time belong to the set ∂u(t, x).In contrast to Definition 1, the notion of viscosity solution [10] for equation (1) on the set Π T requires that this solution satisfies inequality (8) at the boundary points (t, x) ∈ Γ T for all (a, s) ∈ D − u(t, x).