In this note, we consider a bipolar hydrodynamic model (Euler-Poisson system) in one space dimension. Denoting by n_i, j_i, P_i(n_i), i = 1, 2, and E the charge densities, current densities, pressures and electric field, the scaled equations of the hydrodynamic model are given by

n 1 t + j 1 x = 0 , j 1 t + j 1 2 n 1 + P 1 ( n 1 ) x = n 1 E - j 1 τ 1 , n 2 t + j 2 x = 0 , j 2 t + j 2 2 n 2 + P 2 ( n 2 ) x = - n 2 E - j 2 τ 2 , λ 2 E x = n 1 - n 2 .

The positive constants τ_i(i = 1,2) and λ denote the relaxation time and the Debye length, respectively. The relaxation terms describe in a very rough manner the damping effect of a possible neutral background charge. The Debye length is related to the Coulomb screening of the charged particles. The hydrodynamic models are generally used in the description of charged particle fluids. These models take an important place in the fields of applied and computational mathematics. They can be derived from kinetic models by the moment method. For more details on the semiconductor applications, see 12 and on the applications in plasma physics, see 13. To begin with, we assume in the present article that the pressure-density functions satisfy

P 1 ( n ) = P 2 ( n ) = n γ , γ ≥ 1 ,

and set τ₁, τ₂ and λ to be one for simplicity In particular, we note that γ = 1 is an important case in the applications of engineer. Hence, (1.1) can be simplifies as

n 1 t + j 1 x = 0 , j 1 t + j 1 2 n 1 + P ( n 1 ) x = n 1 E - j 1 , n 2 t + j 2 x = 0 j 2 t + j 2 2 n 2 + P ( n 2 ) x = - n 2 E - j 2 , E x = n 1 - n 2 .

Recently, many efforts were made for the bipolar isentropic hydrodynamic equations of semiconductors. More precisely, Zhou and Li 4 and Tsuge 5 discussed the unique existence of the stationary solutions for the one-dimensional bipolar hydrodynamic model with proper boundary conditions. Natalini 6, and Hsiao and Zhang 78 established the global entropic weak solutions in the framework of compensated compactness on the whole real line and spatial bounded domain respectively. Zhu and Hattori 9 proved the stability of steady-state solutions for a recombined bipolar hydrodynamical model. Ali and Jüngel 10 studied the global smooth solutions of Cauchy problem for multidimensional hydrodynamic models for two-carrier plasma. Lattanzio 11 and Li 12 studied the relaxation time limit of the weak solutions and local smooth solutions for Cauchy problems to the bipolar isentropic hydrodynamic models, respectively. Gasser and Marcati 13 discussed the relaxation limit, quasineutral limit and the combined limit of weak solutions for the bipolar Euler-Poisson system. Gasser et al. 14 investigated the large time behavior of solutions of Cauchy problem to the bipolar model basing on the fact that the frictional damping will cause the nonlinear diffusive phenomena of hyperbolic waves, while Huang and Li recently studied large-time behavior and quasineutral limit of L^∞ solution of the Cauchy problem in 15. As far as we know, no results about the global existence and large time behavior to (1.2) with boundary effect can be found. In this article we will consider global existence and asymptotic behavior of smooth solutions to the initial boundary value problems for the bipolar Euler-Poisson system on the quarter plane ℝ₊ × ℝ₊. Then, we now prescribe the initial-boundary value conditions:

n 1 , j 1 , n 2 , j 2 ( x , 0 ) = n 10 , j 10 , n 20 , j 20 ( x ) → n + , j + , n + , j + as x → + ∞ ,

and

j 1 ( 0 , t ) = j 2 ( 0 , t ) = 0 = E ( 0 , t ) .

Moreover, we also investigate the time-asymptotic behavior of the solutions to (1.2)-(1.4). Our results discussed below show that even for the case with boundary condition, the solutions of (1.2)-(1.4) can be captured by the corresponding porous equation as in initial data case. For the sake of simplicity, we can assume j₊ = 0. This assumption can be removed because of the exponential decay of the momentum at x = ±∞ induced by the linear frictional damping.

Finally, set

( φ i 0 , z i 0 ) : = - ∫ x ∞ n i 0 ( y ) - n ̄ i y + x i 0 , 0 d y , j i 0 ( x ) - j ̄ i ( x , 0 ) ,

here the nonlinear diffusion waves n ̄ 1 , j ̄ 1 , n ̄ 2 , j ̄ 2 will be defined in Section 2, and the shift x_i0 satisfy

∫ 0 ∞ n i 0 ( x ) - n ̄ i x + x i 0 , t = 0 d x = 0 ,

which can be computed from the standard arguments, see 16171819.

Throughout this article C always denotes a harmless positive constant. L^p(ℝ₊) is the space of square integrable real valued function defined on ℝ₊ with the norm ⋅ L p ( ℝ + ) and H^k(ℝ₊) denotes the usual Sobolev space with the norm ∥·∥_k.

Now one of main results in this paper is stated as follows.

Theorem 1.1 Suppose that n₁₀-n₊, n₂₀-n₊ ∈ L¹(ℝ⁺) and satisfies (2.4) for some δ₀ > 0, (φ₁₀, z₁₀, φ₂₀, z₂₀) ∈ (H³(ℝ⁺) ∩ L¹(ℝ₊)) × (H²(ℝ₊) ∩ L¹(ℝ₊)) × (H³(ℝ⁺) ∩ L¹(ℝ₊)) × (H²(ℝ₊) ∩ L¹(ℝ₊)) with x₁₀ = x₂₀ and that

n 10 - n + L 1 ( ℝ + ) + n 20 - n + L 1 ( ℝ + ) + φ 10 , φ 20 H 3 ( ℝ + ) + z 10 , z 20 H 2 ( ℝ + ) + φ 10 , φ 20 L 1 ( ℝ + ) + z 10 , z 20 L 1 ( ℝ + ) + δ 0 ≪ 1

hold. Then there exists a unique time-global solution (n₁, j₁, n₂, j₂)(x, t) of IBVP (1.2)-(1.4), such that for i = 1,2,

n i - n ̄ i ∈ C k 0 , ∞ , H 2 - k ℝ + , k = 0 , 1 , 2 , j i = j ̄ i ∈ C k 0 , ∞ , H 1 - k ℝ + , k = 0 , 1 , E ∈ C k 0 , ∞ , H 3 - k ℝ + , k = 0 , 1 , 2 , 3 ,

and

∂ x k n 1 - n ̄ 1 , n 2 - n ̄ 2 L 2 ( ℝ + ) ≤ C ( 1 + t ) - k 2 , k = 0 , 1 , 2 ,

∂ x k j 1 - j ̄ 1 , j 2 - j ̄ 2 L 2 ( ℝ + ) ≤ C ( 1 + t ) - k + 2 2 , k = 0 , 1 , 2 ,

∂ x k E L 2 ( ℝ + ) ≤ C e - α t , k = 0 , 1 , 2 ,

where α > 0 and C is positive constant.

Next, with the help of Fourier analysis, we can obtain the following optimal convergence rate.

Theorem 1.2 Under the assumptions of Theorem 1.1, it holds that

∂ x k n 1 - n ̄ 1 , n 2 - n ̄ 2 L 2 ( ℝ + ) ≤ C ( 1 + t ) - 2 k + 3 4 , k = 0 , 1 ,

n 1 - n ̄ 1 , n 2 - n ̄ 2 L p ( ℝ + ) ≤ C ( 1 + t ) - 2 p - 1 2 p , 2 ≤ p ≤ + ∞ ,

j 1 - j ̄ 1 , j 2 - j ̄ 2 L 2 ( ℝ + ) ≤ C ( 1 + t ) - 5 4 .

Remark 1.3 The condition (2.4) implies

∫ 0 ∞ n 10 ( x ) - n 20 ( x ) d x = 0 ,

and it is a technique one. As to more general case, we will discuss it in the forthcoming future. Theorems 1.1 and 1.2 show that the nonlinear diffusive phenomena is maintained in the bipolar Euler-Poisson system with the interaction of two particles and the additional electric field, which indeed implies that this diffusion effect is essentially due to the friction of momentum relaxation.

Using the energy estimates, we can establish a priori estimate, which together with local existence, leads to global existence of the smooth solutions for IBVP (1.2)-(1.4) by standard continuity arguments. In order to obtain the asymptotic behavior and optimal decay rate, noting that E = φ₁ - φ₂ satisfies the damping "Klein-Gordon" equation (see 1415), we first obtain the exponential decay rate of the electric field E by energy methods. Then, we can establish the algebraical decay rate of the perturbed densities φ₁ and φ₂. Finally, from the estimates of the wave equation with damping in 20 and using the idea of 16, we show the optimal algebraical decay rates of the total perturbed density φ₁ + φ₂, which together with the exponential decay rate of the difference of two perturbed densities, yields the optimal decay rate. In these procedure, we have overcome the difficulty from the coupling and cancelation interaction between n₁ and n₂. Finally, it is worth mentioning that similar results about the Euler equations with damping have been extensively studied by many authors, i.e., the authors of 161718192122, etc.

The rest of this article is arranged as follows. We first construct the optimal nonlinear diffusion waves and recall some inequalities in Section 2. In Section 3, we reformulate the original problem, and show the main Theorem. Section 4 is to prove an important decay estimate, which has been used to show the main theorem in Section 3.

In this section, we first construct the optimal nonlinear diffusion waves of (1.2) in the quarter plane. To begin with, we define our diffusion waves as

n ̄ i = n + + δ 0 ϕ ( x , t + 1 ) , j ̄ 1 = - P n ̄ i x , i = 1 , 2 .

Here the function ϕ(x, t + 1) (here using t + 1 instead of t is to avoid the singularity of solution decay at the point t = 0) solves

δ 0 ϕ t - P n + + δ 0 ϕ x x = 0 , ( x , t ) ∈ ℝ + × ℝ + ,

namely,

ϕ t - P ′ ( n + ) ϕ x x = 1 δ 0 P n + + δ 0 ϕ - P ( n + ) - P ′ ( n + ) ϕ x x , ( x , t ) ∈ ℝ + × ℝ + ,

with the initial boundary values

ϕ x | x = 0 = 0 , ϕ | x = + ∞ = 0 , ϕ | t = 0 = ϕ ( x , 1 ) = : ϕ 0 ( x ) .

Where ϕ₀(x) is a given smooth function such that

ϕ 0 ( x ) ∈ L 1 ( ℝ + ) and ∫ 0 ∞ ϕ 0 ( x ) d x ≠ 0 ,

and δ₀ is a constant satisfying

∫ 0 ∞ n i 0 ( x ) - n + d x - δ 0 ∫ 0 ∞ ϕ 0 ( x ) d x = 0 .

Note that from the assumptions in Theorem 1.1 and (2.3), there exists δ₀ satisfies (2.4).

The existence of ϕ(x, t) has been shown in 16, and the following estimates of ϕ(x, t) hold:

∂ t j ∂ x k ϕ L 2 ( ℝ + ) ≤ C δ 0 ( 1 + t ) - ( 4 j + 2 k + 1 ) / 4 ,

ϕ x t L 1 ( ℝ + ) ≤ C δ 0 ( 1 + t ) - 3 / 2

with the help of the Green function method and energy estimates.

Then (n₁, j₁, n₂, j₂) (x, t) is the required nonlinear diffusion wave, and satisfies

n ̄ i t + j ̄ i x = 0 , j ̄ i = - P ( n ̄ i ) x ,

with the boundary restrictions

n ̄ i x | x = 0 = 0 , n i | x = + ∞ = n + .

From (2.5) and (2.6), we have

Lemma 2.1 If n ̄ i , j ̄ i (x, t) is defined as above, then

∂ t l ∂ x k n ̄ i - n + L 2 ( ℝ + ) ≤ C δ 0 ( 1 + t ) - ( 4 l + 2 k + 1 ) / 4 ,

n ̄ i x t L 1 ( ℝ + ) ≤ C δ 0 ( 1 + t ) - 3 / 2 .

Next, we introduce some inequalities of Sobolev type.

Lemma 2.2 The following inequalities hold

f L p ( ℝ + ) ≤ C f 1 , p ∈ [ 2 , ∞ ]

for some constant C > 0.

Finally, for later use, we also need

Lemma 2.3 20 Assume that K_i(x, t)(i = 0,1) are the fundamental solutions of

K i t t + K i t - K i x x = 0 , i = 0 , 1

with

K 0 ( x , 0 ) = δ ( x ) , K 1 ( x , 0 ) = 0 , d d t K 0 ( x , 0 ) = 0 , d d t K 1 ( x , 0 ) = δ ( x ) ,

where δ(x) is the Delta function.

If f ∈ L¹(ℝ₊) ∩ H^j+k-1(ℝ₊), then

∂ t j ∂ x k ∫ 0 ∞ K 1 ( x - y , t ) - K 1 ( x + y , t ) f ( y ) d y L 2 ( ℝ + ) ≤ C ( 1 + t ) - j - 2 k + 1 4 f L 1 ( ℝ + ) + f j + k - 1 .

If f ∈ L¹(ℝ₊) ∩ H^j+k(ℝ₊), then

∂ t j ∂ x k ∫ 0 ∞ K 0 ( x - y , t ) - K 0 ( x + y , t ) f ( y ) d y L 2 ( ℝ + ) ≤ C ( 1 + t ) - j - 2 k + 1 4 f L 1 ( ℝ + ) + f j + k .

In this section we are going to reformulate the original problem and establish the global existence and algebraical decay rate. To begin with, from (1.2) and (2.7), we notice that

∫ 0 ∞ ( n i - n ̄ i ) ( x , t ) = ∫ 0 ∞ ( n i 0 - n + ) d x - δ 0 ∫ 0 ∞ ϕ 0 ( x ) d x = 0 , i = 1 , 2 .

Thus, it is reasonable to introduce the following perturbations as our new variables

φ i = - ∫ x ∞ ( n i - n ̄ i ) ( y , t ) d y , z i = j i - j ̄ i ,

which yields

φ 1 t + z 1 = 0 , z 1 t + ( z 1 + j ̄ 1 ) 2 n ̄ 1 + φ 1 x + P ( n ̄ 1 + φ 1 x ) - P ( n ̄ 1 ) x + z 1 = ( n ̄ 1 + φ 1 x ) E + P ( n ̄ 1 ) x t , φ 2 t + z 2 = 0 , z 2 t + ( z 2 + j ̄ 2 ) 2 n ̄ 2 + φ 2 x + P ( n ̄ 2 + φ 2 x ) - P ( n ̄ 2 ) x + z 2 = - ( n ̄ 2 + φ 2 x ) E + P ( n ̄ 2 ) x t φ 2 t + z 2 = 0 , E = φ 1 - φ 2 , ( φ 1 , z 1 , φ 2 , z 2 ) t = 0 = ( φ 10 , z 10 , φ 10 , z 10 ) ( x ) , ( φ 1 , φ 2 ) | x = 0 = 0 .

Further, we have

φ 1 t t - P n ̄ 1 + φ 1 x - P ( n ̄ 1 ) x + φ 1 t + ( n ̄ 1 + φ 1 x ) E = P ( n ̄ 1 ) x t + f 1 x ,

φ 2 t t - P ( n ̄ 2 + φ 2 x ) - P ( n ̄ 2 ) x + φ 2 t - n ̄ 2 + φ 2 x E = P ( n ̄ 2 ) x t + f 1 x ,

and

E t t + E t + ( n ̄ 1 + n ̄ 2 ) E = f 3 ,

here

f i = ( φ i t + P ( n ̄ 1 ) x ) 2 n ̄ i + φ i x , i = 1 , 2 , f 3 = - P ( n ̄ 1 ) x t + P ( n ̄ 2 ) x t + P ( n ̄ 1 + φ 1 x ) - P ( n ̄ 1 ) x - P n ̄ 2 + φ 2 x - P ( n ̄ 2 ) x - ( φ 1 x + φ 2 x ) E + f 1 - f 2 x .

Next, by the standard continuous arguments, we can obtain the global existence of smooth solutions. That is, we combine the local existence and a priori estimate. For the local existence of the solution to (3.2)-(3.3), we see, e.g., 20 and references therein. In the following we devote ourselves to the a priori estimates of the solution (φ₁, φ₂, E)(0 < t < T) to (3.2)-(3.3) under the a priori assumption

N ( T ) : = sup 0 < t < T ( 1 + t ) k ∂ x k ( φ 1 , φ 2 ) L 2 ( ℝ + ) 2 + ( 1 + t ) k + 2 ∂ x k ( φ 1 t , φ 2 t ) L 2 ( ℝ + ) 2 ≪ 1 .

Noting

φ i ( 0 , t ) = φ i t ( 0 , t ) = φ i x x ( 0 , t ) = φ i t x x ( 0 , t ) = 0 , i = 1 , 2 ,

we can obtain the following estimates by using a similar argument of 14. Since the proof is tedious but similar as in the previous works, we only list the results and omit its details.

Lemma 3.1 For T > 0, let (φ₁, φ₂, E)(x, t) be the solution to (3.2)-(3.3). Then, it holds for N(T) + δ₀ that

( φ 1 , φ 2 ) 3 2 + ( φ 1 t , φ 2 t , E ) 3 2 + ∫ 0 T ( φ 1 x , φ 2 x , φ 1 t , φ 2 t , E ) 2 2 d t ≤ C φ 10 , φ 20 3 2 + ( z 10 , z 20 ) 2 2 + δ 0 .

Lemma 3.2 For T > 0, let (φ₁, φ₂, E)(x, t) be the solution to (3.2)-(3.3). Then, it holds for N(T) + δ₀ that

E , E x , E t , E x x , E x t , E t t 2 ≤ C φ 10 , φ 20 3 2 + ( z 10 , z 20 ) 2 2 + δ 0 e - β t ,

for some positive constant β.

Lemma 3.3 For T > 0, let (φ₁, φ₂, E)(x, t) be the solution to (3.2)-(3.3). Then there exist positive constants C such that

∂ x k ( φ 1 , φ 2 ) L 2 ( ℝ + ) ≤ C ( 1 + t ) - k 2 , k = 0 , 1 , 2 , 3 ,

∂ x k ( z 1 , z 2 ) L 2 ( ℝ + ) ≤ C ( 1 + t ) - k + 2 2 , k = 0 , 1 , 2 ,

and

( 1 + t ) 2 z 1 t , z 2 t L 2 ( ℝ + ) + ( 1 + t ) - 5 2 z 1 x t , z 2 x t L 2 ( ℝ + ) + z 1 t t , z 2 t t L 2 ( ℝ + ) ≤ C .

In conclusion, we have

Theorem 3.4 Under the assumptions in Theorem 1.1, there exists a unique time global solution (φ₁, z₁, φ₂, z₂) of the IBVP (3.1) such that

φ 1 , φ 2 , E ∈ C k ( 0 , ∞ , H 3 - k ( ℝ + ) ) , k = 0 , 1 , 2 , 3 ,

z 1 , z 2 ∈ C k ( 0 , ∞ , H 2 - k ( ℝ + ) ) , k = 0 , 1 , 2 ,

and there exist positive constants C, α such that

∂xk(φ1,φ2)L2(ℝ+)≤C(1+t)-k2,k=0,1,2,3,

∂ x k ( z 1 , z 2 ) L 2 ( ℝ + ) ≤ C ( 1 + t ) - k + 2 2 , k = 0 , 1 , 2 ,

E , E x , E t , E x t , E x x L 2 ( ℝ + ) ≤ C e - α t ,

and

(1+t)2z1t,z2tL2(ℝ+)+(1+t)-52z1xt,z2xtL2(ℝ+)+z1tt,z2ttL2(ℝ+)≤C.

In this section we are going to show the optimal decay rate. First of all, we improve the decay rates in Theorem 3.4 to be optimal as follows.

Proposition 4.1 Under the assumptions in Theorem 1.1, the solution (φ₁, z₁, φ₂, z₂) decay time asymptotically as

∂ x k ( φ 1 , φ 2 ) L 2 ( ℝ + ) ≤ C ( 1 + t ) - 2 k + 1 4 , k = 0 , 1 , 2 ,

z 1 , z 2 L 2 ( ℝ + ) ≤ C ( 1 + t ) - 5 4 .

Based on the above Proposition, we can immediately prove Theorem 1.2.

Proof of Theorem 1.2 Thanks to Proposition 4.1, and by noting that φ i x = n i - n ̄ i , z i = j i - j ̄ i , we have

∂ x k n 1 - n ̄ 1 , n 2 - n ̄ 2 ( t ) L 2 ( ℝ + ) = ∂ x k + 1 ( φ 1 , φ 2 ) ( t ) L 2 ( ℝ + ) ≤ ∂ x k + 1 ( φ 1 , φ 2 ) ( t ) L 2 ( ℝ + ) ≤ C ( 1 + t ) - 2 ( k + 1 ) + 1 4 , k = 0 , 1 ,

and

j 1 - j ̄ 1 , j 2 - j ̄ 2 ( t ) L 2 ( ℝ + ) = ( z 1 , z 2 ) ( t ) L 2 ( ℝ + ) ≤ C ( 1 + t ) - 5 4 .

This proved (1.8) and (1.10). Next, using Lemma 2.2, we get

n 1 - n ̄ 1 L p ( ℝ + ) ≤ n 1 - n ̄ 1 L ∞ ( ℝ + ) p - 2 p n 1 - n ̄ 1 L 2 ( ℝ + ) 2 p ≤ 2 n 1 - n ̄ 1 L 2 ( ℝ + ) 1 2 ∂ x ( n 1 - n ̄ 1 ) L 2 ( ℝ + ) 1 2 p - 2 p n 1 - n ̄ 1 L 2 ( ℝ + ) 2 p = 2 p - 2 2 p n 1 - n ̄ 1 L 2 ( ℝ + ) p - 2 2 p + 2 p ∂ x ( n 1 - n ̄ 1 ) L 2 ( ℝ + ) p - 2 2 p ≤ C ( 1 + t ) - 3 4 × p - 2 2 p + 2 p ( 1 + t ) - 5 4 × p - 2 2 p ≤ C ( 1 + t ) - 1 - 1 2 p ,

and

n 2 - n ̄ 2 L p ( ℝ + ) ≤ n 2 - n ̄ 2 L ∞ ( ℝ + ) p - 2 p n 2 - n ̄ 2 L 2 ( ℝ + ) 2 p ≤ 2 n 2 - n ̄ 2 L 2 ( ℝ + ) 1 2 ∂ x ( n 2 - n ̄ 2 ) L 2 ( ℝ + ) 1 2 p - 2 p n 2 - n ̄ 2 L 2 ( ℝ + ) 2 p = 2 p - 2 2 p n 2 - n ̄ 2 L 2 ( ℝ + ) p - 2 2 p + 2 p ∂ x ( n 2 - n ̄ 2 ) L 2 ( ℝ + ) p - 2 2 p ≤ C ( 1 + t ) - 3 4 × p - 2 2 p + 2 p ( 1 + t ) - 5 4 × p - 2 2 p ≤ C ( 1 + t ) - 1 - 1 2 p .

This prove (1.9).

In the following we focus on the proof of Proposition 4.1. To begin with, we notice that

( φ 1 + φ 2 ) t t - P ′ ( n + ) ( φ 1 + φ 2 ) x x + ( φ 1 + φ 2 ) t = F ,

with

F = P ( n ̄ 1 + φ 1 x ) - P ( n ̄ 1 ) - P ′ ( n + ) φ 1 x x + P ( n ̄ 2 + φ 2 x ) - P ( n ̄ 2 ) - P ′ ( n + ) φ 2 x x - P ( n ̄ 1 ) x t - P ( n ̄ 2 ) x t - ( φ 1 x + φ 2 x ) E + ( f 1 + f 2 ) x .

Proof of Proposition 4.1. Firstly, we prove the optimal decay rates for ∂ x k ( φ 1 , φ 2 ) L 2 ( k = 0 , 1 , 2 ) , namely, (4.1). By (4.3) and Lemma 2.3, we obtain

( φ 1 + φ 2 ) ( x , t ) = ∫ 0 ∞ K 0 ( x - y , t ) - K 0 ( x + y , t ) ( φ 10 + φ 20 ) ( y ) d y + ∫ 0 ∞ K 1 ( x - y , t ) - K 1 ( x + y , t ) ( z 10 + z 20 ) ( y ) d y + ∫ 0 t ∫ 0 ∞ K 1 ( x - y , t - τ ) - K 1 ( x + y , t - τ ) F ( y , τ ) d y d τ .

By differentiating (4.4) k-times (k = 0, 1, 2) with respect to x, and by taking its L²(ℝ₊)-norm, we obtain

∂ x k ( φ 1 + φ 2 ) ( t ) L 2 ( ℝ + ) = ∂ x k ∫ 0 ∞ K 0 ( x - y , t ) - K 0 ( x + y , t ) ( φ 10 + φ 20 ) ( y ) d y L 2 ( ℝ + ) + ∫ 0 ∞ K 1 ( x - y , t ) - K 1 ( x + y , t ) ( z 10 + z 20 ) ( y ) d y L 2 ( ℝ + ) + ∫ 0 t ∂ x k ∫ 0 ∞ K 1 ( x - y , t - τ ) - K 1 ( x + y , t - τ ) F ( y , τ ) d y L 2 ( ℝ + ) d τ .