Existence of classical solutions for a class of nonlinear impulsive evolution partial differential equations

This paper is devoted to the study of a class of impulsive nonlinear evolution partial differential equations. We give new results about existence and multiplicity of global classical solutions. The method used is based on the use of fixed points for the sum of two operators. Our main results will be illustrated by an application to an impulsive Burgers equation.

boundary value problem for a nonlinear parabolic partial differential equation was discussed in [9].The approximate controllability of an impulsive semilinear heat equation was proved in [1].A class of impulsive wave equations was investigated in [18].In [27] a class of impulsive semilinear evolution equations with delays is investigated for existence and uniqueness of solutions.The investigations in [27] includes several important partial differential equations such as the Burgers equation and the Benjamin-Bona-Mahony equation with impulses, delays and nonlocal conditions.A class of semilinear neutral evolution equations with impulses and nonlocal conditions in a Banach space is investigated in [2] for existence and uniqueness of solutions.To prove the main results in [2] the authors use a Karakostas fixed point theorem.In [2] an example involving Burger's equation is provided to illustrate the application of the main results.Some studies concerning impulsive Burgers equation can be found in [14,25,33].
Many classical methods have been successfully applied for solving impulsive partial differential equations.By using variational method, the existence of solutions for a fourth-order impulsive partial differential equations with periodic boundary conditions was obtained in [28].The Krasnoselskii theorem is used to prove existence and uniqueness of solutions for impulsive Hamilton-Jacobi equation in [34].Some other references on impulsive partial differential equations are: [3,7,11,16,17,[22][23][24]26,32,35,38,41].
In this paper, we investigate the following class of nonlinear impulsive evolution partial differential equations where Note that for ψ(u) = 1 2 u 2 , we get impulsive Burgers equations.Assume that (A1) 0 , . . ., k}, (A4) there exist a positive constant A and a function g ∈ C(J × R) such that g > 0 on (0, T ] × (R\{x = 0}) and In the last section, we will give an example for a function g that satisfies (A4).Assume that the constants B and A which appear in the conditions (A1) and (A4), respectively, satisfy the following inequalities: (A5) AB 1 < B, where and L is a positive constant that satisfies the following conditions: with r and R 1 are positive constants and m is the constant which appear in (A6).
Our aim in this paper is to investigate the problem (1.1) for existence of classical solutions.Let J 0 = J \{t j } k j=1 and define the spaces PC(J ), PC 1 (J ) and PC 1 (J, C 1 (R n )) by PC(J ) = {g : g ∈ C(J 0 ), ∃g(t + j ), g(t − j ) and g(t − j ) = g(t j ), j ∈ {1, . . ., k}}, PC 1 (J ) = {g : g ∈ PC(J ) ∩ C 1 (J 0 ), ∃g (t − j ), g (t + j ) and g (t − j ) = g (t j ), j ∈ {1, . . ., k}} and Our main results are as follows. Theorem Our work is motivated by the interest of researchers for many mathematical questions related to impulsive partial differential equations.In fact, some important applied problems reduce to the study of such equations, see for example, [6,13,20,23,29,36,43].Some applications of the impulsive PDEs in the quantum mechanics can be found in [36].The asymptotical synchronization of coupled nonlinear impulsive partial differential systems in complex networks was considered in [43].Applications are given to models in ecology in [20].
Applications to the population dynamics are given in [6,13,23].A cell population model described by impulsive PDEs was studied in [29].This paper is organized as follows.In the next section, we give some existence and multiplicity results about fixed points of the sum of two operators.Then in Sect.3, we prove our main results.First, we give an integral representation and a priori estimates related to solutions of problem (1.1).Then, we use these estimates to prove Theorems 1.1 and 1.2 by using the results on the sum of operators recalled in Sect. 2. Finally, in Sect.4, we illustrate our main results by an application to an impulsive Burgers equation.

Fixed points for the sum of two operators
The following theorem concerns the existence of fixed points for the sum of two operators.Its proof can be found in [18].

Theorem 2.1 Let E be a Banach space and
with R > 0. Consider two operators T and S, where with > 0 and S : E 1 → E be continuous and such that In the sequel, E is a real Banach space.Definition 2.2 A closed, convex set P in E is said to be cone if (i) αx ∈ P for any α ≥ 0 and for any x ∈ P, (ii) x, −x ∈ P implies x = 0. Definition 2. 4 Let X and Y be real Banach spaces.A mapping K : X → Y is said to be expansive if there exists a constant h > 1 such that The following theorem concerns the existence of nonnegative fixed points for the sum of two operators.The details of its proof can be found in [12] and [31].Theorem 2.5 Let P be a cone of a Banach space E; a subset of P and U 1 , U 2 and U 3 three open bounded subsets of P such that U 1 ⊂ U 2 ⊂ U 3 and 0 ∈ U 1 .Assume that T : → P is an expansive mapping, S : U 3 → E is a completely continuous and S(U 3 ) ⊂ (I −T )( ).Suppose that (U 2 \U 1 )∩ = ∅, (U 3 \U 2 )∩ = ∅, and there exists w 0 ∈ P\{0} such that the following conditions hold: Then T + S has at least two non-zero fixed points x 1 , x 2 ∈ P such that 3 Proof of the main results

Integral representation and a priori estimates related to solutions of problem (1.1)
In the sequel, we will denote the space PC 1 (J, C 1 (R)) defined in (1.2) by X and it will be endowed by the following norm: respectively, (ii) The estimation of |I j (x, u(t, x))|, (t, x) ∈ J × R, j ∈ {1, . . ., k} : (iii) The estimation of This completes the proof.
For u ∈ X , define the operator then it is a solution to the IVP (1.1).

Proof
We have Hence, We differentiate (3.2) with respect to t and we find We put t = 0 in (3.2) and we get u(0, x) = u 0 (x), x ∈ R. Now, by (3.2), we obtain We apply Lemma 3.1 and we get This completes the proof.For u ∈ X , define the operator with g is the function which appears in the condition (A4).Lemma 3.4 Suppose (A1)-(A4).If u ∈ X and u ≤ B, then where Thus, S 2 u ≤ B. This completes the proof.
Proof We differentiate two times with respect to t and two times with respect to x the Eq.(3.4) and we find Thus, Hence and Lemma 3.2, we conclude that u is a solution to the IVP (1.1).This completes the proof.

Proof of Theorem 1.1
Suppose that the constants B, A and B 1 are those which appear in the conditions (A1), (A4) and (A5), respectively.Choose ∈ (0, 1), such that By the Ascoli-Arzelà theorem, it follows that Y is a compact set in X .For u ∈ X , define the operators where S 2 is the operator defined by formula (3.3).For u ∈ Y , using Lemma 3.4, we have Thus, S : Y → X is continuous and which is a contradiction.Hence and Theorem 2.1, it follows that the operator T + S has a fixed point u * ∈ Y .Therefore From here and from Lemma 3.5, it follows that u is a solution to the problem (1.1).This completes the proof.
With P we will denote the set of all equi-continuous families in P. For v ∈ X , define the operators where is a positive constant, m > 0 is the constant which appear in (A6) and the operator S 2 is given by formula (3.3).Note that any fixed point v ∈ X of the operator T 1 + S 3 is a solution to the IVP (1.1).Now, let us define where r, L , R 1 , A, B 1 are the constants which appear in condition (A6).
2. For v ∈ P R 1 , we get Therefore S 3 (P R 1 ) is uniformly bounded.Since S 3 : P R 1 → X is continuous, we have that S 3 (P R 1 ) is equi-continuous.Consequently S 3 : P R 1 → X is completely continuous.

Let ε
which is a contradiction.
Therefore all conditions of Theorem 2.5 hold.Hence, the problem (1.1) has at least two solutions u 1 and u 2 so that

An Example
Below, we will illustrate our main results.Let k = 2, and Then and i.e., (A5) holds.Next, i