SOLUTIONS TO VISCOUS BURGERS EQUATIONS WITH TIME DEPENDENT SOURCE TERM

. We study the existence and uniqueness of weak solutions for a Cauchy problem of a viscous Burgers equation with a time dependent reaction term involving Dirac measure. After applying a Hopf like transformation, we investigate the associated two initial boundary value problems by assuming a common boundary. The existence of the boundary data is shown with the help of Abel’s integral equation. We then derive explicit representation of the boundary function. Also, we prove that the solutions of associated initial boundary value problems converge uniformly to a nonzero constant on compact sets as t approaches ∞


Introduction
This article concerns the existence, uniqueness and regularity of solutions to the Burgers equation with time dependent point source, u t + uu x − u xx = 2 1 + t δ(x), x ∈ R, t > 0, (1.1) subject to the initial condition u(x, 0) = u 0 (x), x ∈ R, (1.2) where u 0 ∈ W 1,∞ (R) ∩ C 1 (R) ∩ L 1 (R). In the literature, the study on viscous Burgers equation with source terms u t + u u x = u xx + f (x, t), x ∈ R, t > 0, (1.3) has obtained much recognition because of extended importance in numerous fields of science, technology and biology [1,17]. Construction of explicit solutions for viscous Burgers equation with inhomogeneous terms and large time behavior of these solutions are discussed by several authors. Eule and Friedrich [8] discussed the solutions of externally forced Burgers equation where H n is Hermite polynomial of order n. An interesting feature of the solutions to (1.7) is that the set of functions {v n (x, 0)} is a complete orthonormal system for the Hilbert space L 2 (R, e 2 ) can be expanded as an infinite series in terms of {v n (x, 0)}. Hence, the solution of (1.5)-(1.6) is represented as an infinite series of self-similar solutions (1.7). Another feature of the constructed self similar solutions (1.7) is that the decay rate is obtained easily which, in turn, gives the large time asymptotes to the solution of (1.5)-(1.6). Ding-Jiu-He [6] constructed the explicit solutions of non-homogeneous Burgers equation u t + u u x = µu xx + kx, x ∈ R, t > 0, (1.8) subject to the initial data u(x, 0) = u 0 (x), x ∈ R, (1.9) where µ > 0 and k is constant. The authors imposed two conditions on the initial function u 0 that it is locally integrable and x 0 u 0 (y) dy = o(x 2 ) as |x| → ∞. They applied Hopf transformation to reduce (1.8)-(1.9) to the linear differential equation and then represented the solution of resulting linear differential equation in terms of Fourier-Hermite series. They proved that the solution u(x, t) of the initial value problem (1.8)-(1.9) behaves like √ kx for large time t. However, Chidella and Yadav [4] noticed that bounded and compactly supported initial functions u 0 do not satisfy the conditions imposed by Ding-Jiu-He [6] and so considered the nonhomogeneous Burgers equation (1.8)-(1.9) with an assumption on u 0 that Using the Hopf transformation and standard transformations of Polyanin and Nazaikinskii [12], they reduced the initial value problem (1.8)-(1.9) to an initial value problem for heat equation and then used the results of Kloosterziel [11] to express the solution of the heat equation in terms of the self-similar solutions of the heat equation. Buyukasik and Pashaev [3] discussed the shock wave solutions, triangular wave solutions, N-wave solutions and rational function solutions for the Burgers equation (1.8). Rao and Yadav [15] considered a non-homogeneous Burgers equation (1.10) subject to the unbounded initial data and expressed the solutions in terms of the self similar solutions of a linear partial differential equation with variable coefficients. They obtained the large time behavior of the solution of the nonhomogeneous Burgers equation. Rao and Yadav [14] investigated solutions for (1.10) by assuming that the initial data is compactly supported and bounded. Engu-Ahmed-Murugan [7] proved the existence of a solution for the initial value problem of a nonhomogeneous Burgers equation and expressed the solution in terms of Hermite polynomials. Their analysis mainly depends on Hopf-Cole transformation and method of variation of parameters. The authors have also given the rates of convergence of an approximate solution to the true solution of the initial value problem. In regards to construction of fundamental solutions of evolutionary equations, we refer to Pskhu [13] and the references there in. However, investigating the solutions for viscous Burgers equation with source term involving the Dirac delta measure becomes complicated as the linearization process of the viscous Burgers equation with the source term leads to two different linear partial differential equations on the two upper quarter planes. Further, considering the nontrivial initial condition with the nonhomogeneous viscous Burgers equation increases the complexity more. Chung-Kim-Slemrod [5] studied the existence, uniqueness and asymptotic behavior of solutions to a Cauchy problem for the viscous Burgers equation with Dirac delta measure as source term.
To understand the viscous Burgers equation with time dependent point source involving dirac function, we consider an initial value problem for the non homogeneous viscous Burgers equation (1.2) where δ is the Dirac delta function concentrated at x = 0. We use the Hopf transformation for linearization. Linearized partial differential equation consists of Heaviside function and so one needs to study the problem on two upper quarter planes separately with common boundary along the positive t-axis. With the help of Abel's integral equation, we intend to establish the existence and uniqueness of the common boundary data of the linearized partial differential equations. We then look for the explicit representation of the boundary function. In view of the integrals involved in representation of the boundary function, we seek the asymptotic behavior of it for large time t. Using this asymptotic behavior, asymptotic behavior of the solution of the linearized partial differential equation is established. Making use of inverse Hopf-Cole tranformation, existence, uniqueness and regularity of the solutions to the non homogeneous viscous Burgers equation is discussed. Eventually, the convergence of the solutions to zero on compact intervals is obtained.
The rest of this article is organized as follows. Section 2 deals with the linearization of the Cauchy problem and then the existence, uniqueness of the common boundary data of the resulting two initial-boundary value problems. This section also discusses the explicit representation of the common boundary data and its asymptotic behavior. In section 3, existence and uniqueness of the global weak solutions for the Cauchy problem is discussed. Further, asymptotic behavior the solutions is investigated.

Burgers equation with inhomogeneous term
Consider the Burgers equation with time dependent point source given by ( Hopf-Cole transformation [2,10], is given by where H(x) is the Heaviside function. The above Cauchy problem is split into two problems, namely, It is to be noted that the same boundary function g(t) is taken for both the initialboundary value problems (2.4) and (2.5) to assume that θ(x, t) is continuous on the positive t axis. Further, we assume temporarily that g(t) is continuously differentiable on [0, ∞) and will be calculated after showing the existence of g(t). Consider the transformation [12] w(x, t) = (1 + t)R(x, t). (2.6) Then the initial-boundary value problem (2.5) reduces to the associated initialboundary value problem for the heat equation Solving (2.7) and retracing R(x, t) using (2.6), we obtain where erfc(x) is the complementary error function Similarly, solving the initial-boundary value problem (2.4), we obtain We impose a constraint on θ(x, t) that space derivatives exist along the positive t-axis and are equal. That is, Calculating R x (x, t) and L x (x, t) from (2.8) and (2.9) respectively and putting x = 0, we find that Hence, using (2.10), we obtain the integral equation (2.11) In the view of the above integral equation, the following theorem discusses the existence and uniqueness of boundary condition g(t).
Proof. The integral equation (2.11) can be written in Abel's integral equation form, and It can be observed that K(t, τ ) is continuous and Note that F (0) cannot be obtained directly from (2.13). Hence, integrating by parts in (2.13) simplifies to for the choice of θ 0 (0) = g(0). Then we notice that F (0) = 0 and The condition on θ 0 and simplification of the above equation lead to |F (t)| ≤ m √ t for some m > 0 depending on θ 0 . We define which yields i.e., D 1/2 F is continuous and bounded for all t > 0. Using standard results [9], there exists a unique continuous bounded solution υ(t) for Abel's equation (2.12). By defining g(t) := θ 0 (0) + t 0 υ(τ )dτ , we conclude that g(t) satisfies all the desired properties in the statement.
Therefore, the solution of (2.2)-(2.3) is well defined and is given by where R(x, t) and L(x, t) are given in (2.8) and (2.9) respectively. Using the results given in [9], we state the bounds for υ as follows, where M = sup t≤τ ∂K ∂t (t, τ ) and for some number γ > 0. Having shown the existence and uniqueness for g(t), we derive explicit expression for it.
Explicit representation of g(t). By reorganizing Abel's integral equation (2.12) and then applying integration by parts lead to the classical Abel's integral equation form [9], Multiplying both sides of (2.17) by where 0 < t < y < ∞ and integrating from 0 to y, we obtain Changing the order of integration and rearranging yields Simplifying the integral in the left side of the above equation directs to Differentiating with respect to t, we obtain Substitution of υ(t) = g (t) and simplification leads to Using F (t) given in (2.13), we evaluate the second term on the right side of the above equation which yields An example. Consider the Cauchy problem (1.1)-(1.2) with trivial initial data u 0 ≡ 0 to have a look at the large time behavior of the boundary condition g(t) easily. In this case, we obtain θ 0 ≡ 1 and B(t) is given by

Integrating the above equation leads to
Hence, equation (2.21) reduces to .
Considering the Cauchy problem (1.1)-(1.2), it is observed that the large time behavior of g(t) will remain same as that of g(t) concerned to the trivial initial data case.
Asymptotic behavior of g(t). Since u 0 ∈ L 1 (R), we have for some real constant k. It is easy to observe that θ 0 (x) → 1 as x → −∞. Proof. Let τ = γt in B(t) given in (2.21). Then Since u 0 is continuous and essentially bounded, there exists a real M > 0 such that Using the above values, we obtain It can be seen that Using these estimates in (2.21) as t → ∞, we complete the proof. (0, ∞)) and u satisfies (1.1) in the sense of distributions. i.e.,
Proof. In view of the fact that u 0 ∈ L 1 (R) and (2.1), we obtain θ 0 (x) > 0 for all x ∈ R. Considering the maximum principle, it is sufficient to show that θ(0, t) = g(t) > 0, ∀ t > 0. On the contrary, assume that θ(0, t) = g(t) ≤ 0 for some t > 0. This implies that there exists a point q > 0 which satisfies g(t) and g(σ) is non-negative for σ ≤ q. By rearranging the kernel in (2.11), Splitting the right side of (3.2) and substituting above we obtain It is clear that the first term of the above equation admits a non-negative lower bound. Using 0 ≤ σ ≤ q < t, we obtain Hence, it is proved that left side of (3.3) admits a positive lower bound as t → q, whereas the right side vanishes as t → q, which is a contradiction.
Proof. Let us prove the existence of a weak solution of (1.1)-(1.2). It is known that , for x ≥ 0.
Hence, we obtain Thus using the above equations one can obtain Let U and V be the domains of R(x, t) and L(x, t) respectively. Integrating u φ t by parts gives Similarly, integrating u xx φ by parts, we obtain Further, integrating u 2 2 φ x by parts on U and V provides (3.6) The dominated convergence theorem and then integration by parts reduce the right hand side expression of above equation to zero. Let us prove uniqueness. Let u and v be two solutions of (3.1), and put w = u−v. Then, we obtain Note that the defined function φ in both domains is not a test function. However, we can use this φ in (3.7) using usual approximation techniques as ). For 0 ≤ x < ∞, the weak derivatives of φ with respect to t is w t (x, t)−w(x, t)δ(t− T ) and the weak derivative of φ with respect to x is w x (x, t)+w(x, t)δ(x). Similarly for −∞ < x < 0, the weak derivatives of φ with respect to t is w t (x, t)−w(x, t)δ(t− T ) and the weak derivative of φ with respect to x is w x (x, t) − w(x, t)δ(x).
Proof. First, we prove that R(x, t) converges to 1/3 uniformly on compact sets. Assume 0 ≤ x ≤ A for some A > 0. Let g(t) be bounded by M . Integrating the second term in (2.8) by parts, we have It is seen that the first term vanishes uniformly as t → ∞ and hence ignore it. Then we consider Expanding the second term of (3.12) and changing the variable, the above expression turns out to be (1 + t) √ τ dτ. (3.13) Note that the second and third term of the above equation admits a uniform bound and vanishes uniformly as t tends to ∞. i.e., one can obtain Since t ≤ 1 + t for all t > 0 and g(t) is bounded, we can see that the last term in (3.13) is uniformly bounded by Now, we consider the uniform convergence of xR x (x, t). We have xR x (x, t) = x (1 + t) which vanishes uniformly. The bound for the second term in right side of (3.20) is obtained as follows: It is observed that the second term in the above expression vanishes uniformly and is bounded by This completes the proof.
Proof. Using the Theorem 3.2, the inverse Hopf-Cole transformation is well defined and given by (3.4). Hence, for x > 0, we have u(x, t) = −2 x R x (x, t) x R(x, t) → 0 as t → ∞.
In view of the uniform convergence of R, R x , L, L x , the unique weak solution u(x, t) converges to zero uniformly on compact sets.