Green’s function of the Cauchy problem is constructed by the method of successive approximations, and its main properties are studied for a new class of linear differential equations with dissipative parabolicity and negative genus, whose coefficients are bounded, continuous in time, and infinitely differentiable by the spatial variable of the function. This class covers Shilov parabolic equations, as well as the class of the Zhitomirskii parabolic Shilov-type equations with variable coefficients and negative genus, and successfully complements the Petrovsky class of parabolic equations.

1. Introduction

Unlike the 2b-parabolic Petrovsky equations with partial derivatives, in Shilov parabolic equations, the order p may no longer coincide with the parabolicity index h, which causes the effect of “parabolic dissipation” measured by special characteristics of the equation—its genus μ [1, 2]: 1 − (p − h) ≤ μ ≤ 1.

Parabolic equations in which p = h, in particular, the classical heat conduction equation and all 2b-parabolic equations, have genus μ = 1, while for equations with p ≠ h in general—genus μ < 1. The more the parabolic index h deviates from the order of the equation p, the more its genus μ decreases and moves away from 1. In equations with such dissipation, even with constant coefficients, deviations from the standards set by the classical heat conduction equation are observed. First of all, for their fundamental solution

, the properties of analyticity in the complex space

deteriorate, and also the order of exponential behavior on the real hyperplane

changes [2, 3]:

()

Another anomalous phenomenon of equations with dissipative parabolicity is their parabolic instability in relation to changing coefficients, even those with zero derivative. Hou-Sin noted this in the article [4]; here, he also gives an example of a parabolically unstable system.

The study of the Cauchy problem for the Shilov parabolic equations was carried out in many research papers. In particular, the classes of uniqueness and correctness of this problem are described in [2]. Properties of the solutions’ stabilization under special Λ-conditions were studied in [5, 6]. In [7], the principle of solution localization on the initial hyperplane is established. Abstract theory of the Cauchy problem in Banach spaces is developed in [8, 9]. The question of correct solvability of the Cauchy problem for equations with inhomogeneity is studied in [10]. The nonlocal Cauchy problem with impulse action is studied in [11, 12].

The presence of parabolic dissipation in the equation significantly complicates the process of its study, particularly in spreading a number of well-known results of the classical theory of the Cauchy problem for Petrovsky parabolic equations [13–15]. The problem of developing the classical theory of boundary value problems, in particular, the Cauchy problem for equations with variable coefficients, is still open. The first results in this direction were obtained by Zhitomirskii back in the 1950s [3], who singled out a class of equations that are parabolically resistant to the changes in their coefficients and established the correct solvability of the Cauchy problem in the class of bounded equations with finite smoothness functions. The Zhitomirskii class generalizes and extends the Shilov class of equations with constant coefficients; however, Zhitomirskii’s research was conducted only in the case of time-independent coefficients, while the solution of the Cauchy problem was constructed by the method of successive approximations without using Green’s function of the problem. Note that the presence of Green’s function enables to obtain more general results, so the construction of this function is important.

This paper deals with a general class of linear differential equations with dissipative parabolicity, negative genus, and smooth bounded variable coefficients, which naturally extends the well-known Zhitomirskii class and generalizes the Shilov class. For this class, Green’s function of the Cauchy problem (hereinafter GFCP) is constructed by the method of successive approximations and its properties are studied based on it. The obtained results are important for the development of the classical theory of the Cauchy problem for equations with dissipative parabolicity and variable coefficients.

The method of researching GFCP for equations with variable coefficients is developed in [16–18].

2. Preliminary Information

Suppose is the real space of dimension n with a scalar product (⋅, ⋅) and the norm , is the complex space, and is the set of all n-dimensional multi-indices; , l > 0, ; λ≔1/(1 − μ/h) and γ≔1/(h − μ). We will also need sets and .

Let us fix an arbitrarily real number T > 0 and consider a differential equation of order p:

()

in which u is an unknown function, i is the imaginary unit, ∂_ξ is the partial derivative with respect to ξ, and

()

are differential expressions of the p and p₁ orders, respectively: 0 ≤ p₁ < p. At the same time, we assume that the differential expression A₀(t; i∂_x) on the set Π_{[0; T]} is {p, h}-parabolic with negative genus μ. This means that the corresponding differential equation

()

is Shilov parabolic on Π_{[0; T]} with the parabolic index h, 0 < h ≤ p, and genus μ < 0.

In addition, we also assume that:

(1)
Coefficients a_0,k(t) and a_1,k(x; t) of equation (2) on the set Π_{[0; T]} are complex-valued functions that are continuous by the variable t, infinitely differentiable by the variable x, and are bounded together with their derivatives.
(2)
The order p₁ of the group of junior members of equation (2) is less than h: p₁ < h.

It should be mentioned that (4) on the set Π_{[0; T]} is called a Shilov parabolic equation, or {p, h}-parabolic one, if

()

with some constants δ₀ > 0 and δ ≥ 0. According to theorems of the Phragmén–Lindel

f type [19], for {p, h}-parabolic equation (4), there is a domain

()

in which

()

Here ν is some number with [1 − (p − h); 1] and C is a positive constant. For {p, h}-parabolic equation (4), the genus μ is the exact upper limit of the indices ν with which the estimate (7) is performed in the domain

The GFCP for the {p, h}-parabolic equation (4) is denoted by

()

where F is the Fourier transform and

()

Let us formulate the following statement.

Theorem 1 (see [20].)Let equation (4) be an {p, h}-parabolic one with genus μ < 0, and let α ≥ 0 and l ≥ 0 be such randomly fixed numbers that (αh − l)μ ≥ αh and l ≤ 1 + αh; then

()

It should be noted that Zhitomirskii estimates (1) for the case μ < 0 follow directly from (10) for α = 0, q = 0, and l = 0.

After putting q = 0, (αh − l)μ = αh, and l = 1 + αh and taking into account the relation

()

from Theorem 1, we obtain the following statement.

Corollary 2 (see [20].)For {p, h}-parabolic equation (4) with genus μ < 0, there exist such positive constants c, B, and δ that for all , , τ ∈ [0; T), and t ∈ (τ; T] the estimate

()

is fulfilled.

Definition 3 (see [21].)We call function Z defined on the set a GFCP for equation (3) if Z as a function (x; t) satisfies equation (3) on the set Π_{(τ; T]}, τ ∈ [0; T), and the boundary relation

()

The task is to construct the function Z for equation (3) and study its main properties. For that, the following estimates [15] will be needed:

()

These estimates are performed for all , 0 ≤ τ < β < t ≤ T, and ε ∈ (0; 1), while the value of c_ε depends only on ε.

3. Method of Successive Approximations

Estimates of the derivatives of the function obtained in [20] allow us to construct the GFCP for equation (2) with genus μ < 0 according to the scheme developed in [21] for the case of nonnegative genus μ.

For equation (2), the GFCP in the form

()

will be constructed, where

is the GFCP for equation (4), while

()

is the volume potential with a certain Φ density. The density Φ should be such that the function Z with respect to the variables x, t is the usual solution of equation (2). Then Φ must be the solution of the following integral equation [21]:

()

where

()

Applying the method of successive approximations to this equation, we obtain the following formal solution:

()

where K₁ = K and

()

To determine the convergence of the series (20) and to substantiate the correctness of the previously performed transformations, first the properties of repeated kernels K_l should be studied.

Since

()

where

is a binomial coefficient, then using the boundedness of derivatives

and estimate (12), we obtain the following estimate:

()

(here c_q,r, δ are independent of x, t, ξ, τ, in addition δ is also independent of r and q).

If l > 1, we will use the obvious equality

()

according to which

()

Thus, the evaluation of

is reduced to the estimation of expressions

()

Taking into account the limitation of the derivative coefficients of equation (2) and estimate (12), for all

, and τ ∈ (0; T], we obtain the following estimates:

()

Now the expression is to be estimated.

Considering the equality

, at l = 1, we find the following:

()

In the integral of the equality

()

we will replace the integration variable according to the rule y = ξ + z. Then taking into account estimates (29) and (15) and the equality

()

for all

, we will get the following estimates:

()

where B(⋅, ⋅) is Euler’s beta function, and γ₀≔1 − p₁/h > 0 (because p₁ < h). Here we used equality

()

Continuing similar considerations, we will arrive at estimates

()

They are performed for all ε ∈ (0; 1),

, τ ∈ (0; T], t ∈ (τ; T], and l ≥ 2. Based on this, we observe the existence of such a number l_∗, for which the following inequality is correct:

()

Since

()

the expressions

and

are of the same type. Therefore, taking into account representation (25), as well as the obtained estimates (27), (28), (34), and (15), for all

, we get the following ratios:

()

Continuing the estimation process by analogy, we make sure that the following estimate is correct:

()

where ε ∈ (0; 1),

, τ ∈ (0; T], t ∈ (τ; T], and l ≥ 2.

Now we continue finding estimates of the expression suitable for establishing the differentiability of the function Φ by spatial variables.

Let l^∗ be the number at which the inequality holds:

()

Putting here l_°≔min{l_∗, l^∗}, l^°≔max{l_∗, l^∗}, where l_∗ is the corresponding number from (35), T₀≔max{1, T}, ε≔1/(l^°r_∗), δ_∗≔δ(1 − 1/r_∗), r_∗ > 2, and also

()

from (34) and (38), we obtain the following estimates:

()

These estimates are correct for all , , t ∈ (τ; T], and τ ∈ (0; T].

Therefore, given the estimate (14), we find the following. Hence, taking into account estimate (14) and the equality

()

as well as considering formula (25) and inequalities (27) and (28), we obtain the following:

()

Thinking similarly, we first make sure of the correctness of the estimates

()

and then of the estimates

()

for

, and

The fixed estimates of the derivatives of repeated kernels enable to formulate the following auxiliary statement.

Lemma 4. The functional series (20) absolutely coincides on the set . Its sum Φ(x, t; ξ, τ) on this set is an infinitely differentiable function for spatial variables x and ξ for which the following estimates are correct:

()

where

, τ ∈ (0; T], t ∈ (τ; T],

, and estimated values c₁, c₂, δ_∗ are independent of t, τ, x, ξ, η, while δ_∗ is also independent of r and q.

Proof. Using estimates (38), (41), and (45), as well as the equality

()

in which Γ(⋅) is Euler’s gamma function, for

, we find the following relation:

()

Hence, we get the absolute convergence of the series (20) on the set .

Now we shall arbitrarily fix the point (x₀; ξ₀) from and consider the sphere of radius δ > 0 centered at this point. For the infinite differentiability of the function Φ at the point (x₀; ξ₀), with fixed t ∈ (τ; T] and τ ∈ (0; T], it is sufficient to establish uniform convergence in of the formally differentiated series (20): . However, this convergence becomes obvious if we consider estimates (38), (41), and (45).

Then, using estimates (38), (41), and (45) once again, we obtain the following estimates for and :

()

Therefore, the fulfillment of estimate (46) is established.

In a similar way, using the corresponding estimates (34), (41), and (44), we assure ourselves of the correctness of estimate (47).

Corollary 5. The integral equation (18) has the solution Φ.

We also note that the statements of Lemma 4 together with estimate (12) provide for all the absolute convergence of the integral that defines the potential W from equality (16). Thus, the function Z(x, t; ξ, τ) is correctly defined by formula (16) on the entire set .

The following statement is correct.

Theorem 6. The function Z(x, t; ξ, τ) on the set is infinitely differentiable by spatial variables ξ and x, and the following holds for its derivatives:

()

Proof. The problem of differentiability of the function Z by spatial variables is a problem about the possibility of differentiation by these variables under the sign of the integral in the volume potential W. To ensure this possibility, it is enough to establish uniform convergence of the integral

()

with respect to the variables x and ξ on

Using the relation

()

as well as estimates (12), (15), (46), and (47), we obtain the following inequality:

()

for all

and

. Here, the value of c₃ does not depend on t, τ, ξ, and x.

This estimate ensures the uniform convergence of the integral I_∗ with respect to the variables x and ξ, and hence the infinite differentiability by these variables of the function Z on the set .

As for statement (51), it becomes obvious if we take into account estimates (12) and (54) and the inequality

()

A simple analysis of the proof of the previous theorem convinces of the correctness of such a conclusion.

Corollary 7. The equality

()

is correct for all τ ∈ [0; T) and t ∈ (τ; T], well as

4. Properties of the Volume Potential in relation to the Time Variable

The following auxiliary statements are important when studying the properties of the function Z(x, t; ξ, τ) with respect to the variables t and τ.

Lemma 8. The function Φ(x, t; ξ, τ) is uniformly continuous with respect to by the variable t on the set (τ; T] for each fixed τ ∈ (0; T], and the following inequality is correct:

()

where ω(⋅) is a positive infinitely small value at zero, independent of x, ξ, t, and τ, while δ₀ is also independent of ζ.

Proof. It is enough to substantiate the correctness of the estimate (57), from which the indicated uniform continuity of the function Φ becomes obvious.

Taking into account the absolute convergence of the series (20), for , t > τ, we obtain the following relation:

()

The common term of the series from the previous inequality should be estimated. We denote by ω₀(⋅) the common modulus of continuity for the variable t of the coefficients of equation (2) and set

()

Further, we shall use the following relations:

()

They are correct for all 0 ≤ τ < t ≤ T, 0 < |ζ| < (t − τ)/2 and θ ∈ (0; 1].

We shall first consider the case ζ > 0. Directly from the differentiability by t of the function and Lagrange mean value theorem (intermediate point theorem), we arrive at the following representation:

()

Hence, taking into account the boundedness and continuity of the coefficients of equation (2) and inequality (12) and the fact that is a solution of equation (4), for 0 < ζ < (t − τ)/2, , we obtain the following estimate:

()

Next, taking into account the structure of repeated kernels K_l, we obtain the inequality

()

in which

()

Using estimates (15) and (38), we find that

()

where ε ∈ (0; 1), 0 < ζ < (t − τ)/2, and

According to the representation (61) and the inequalities (12), (15), and (38), we obtain estimates

()

in which {θ, ε} ⊂ (0; 1),

, and 0 < ζ < (t − τ)/2.

Therefore, there exists such a positive constant δ that for and ε ∈ (0; 1) there is a value with which for all 0 < ζ < (t − τ)/2 and the estimate

()

is fulfilled.

This estimate ensures the existence of such positive constants c₀, δ₀ and numbers l₀ that for and 0 < ζ < (t − τ)/2, the following inequality holds true:

()

Hence, using (12), (62), and (63), as well as the limitation and continuity of the coefficients of equation (2), and thinking in a similar way to obtaining the estimate (45), we arrive at the following estimate:

()

Here

and 0 < ζ < (t − τ)/2, while the estimated values δ and c do not depend on x, ξ, t, τ, and l.

In conclusion, we note that the obtained estimates (68)–(70) for a positive ζ ensure the fulfillment of inequality (57).

In the case of ζ < 0, inequality (57) is established in a similar way.

Lemma 9. The function

()

is continuously differentiable by the variable t on the set (β; T] and continuous by the variable β on the interval (τ; t), and besides the boundary relation

()

is satisfied.

Proof. Directly from estimate

()

which holds for all

and β ∈ (τ; t), 0 ≤ τ < t ≤ T, we arrive at the differentiability of

by the variable t at each point of the interval (β; T] and the fulfillment of equality

()

Using the fact that the function is a solution of equation (4), the limitation and continuity of the coefficients of this equation, and Lagrange mean value theorem and the formula for integration by parts, we obtain the following estimate:

()

Hence, taking into account the inequalities (12), (15), and (46), for 0 < |ζ| < (t − β)/2, , and β ∈ (τ; t), 0 ≤ τ < t ≤ T, we obtain the estimate

()

in which the positive values c and δ do not depend on x, ξ, t, τ, β, and ζ.

Therefore, the function is continuously differentiable by the variable t on the set (β; T] for each fixed τ ∈ [0; T), β ∈ (τ; T), x, and .

Further, using the representation

()

the limitation of the coefficients of equation (4), and estimates (12), (46), and (57), we find the following estimates:

()

for all

, 0 < |ζ| < min{(t − β), (β − τ)}/2, θ ∈ (0; 1), β ∈ (τ; t), 0 ≤ τ < t ≤ T (here c₁ > 0 does not depend on ζ).

These estimates guarantee the continuity of the function with respect to the parameter β on the interval (τ; t).

We should now prove the fulfillment of the limit equality (72). For this, we will use the fact that belongs to the space as a convolution of the elements of this space because and

()

where

is the Schwartz space of basis functions and “ ∗” is the convolution operation in

Let

()

Taking into account the equality and the properties of the operator F, we come to the conclusion that to prove the boundary relation (72), it is sufficient to determine the fulfillment of the relation

()

in the sense of convergence in the space

, i.e., to make sure of the correctness of such statements [19]:

(I) For each fixed , t ∈ (τ; T], and τ ∈ [0; T), the relation
()
is fulfilled (here we are talking about a uniform direction with respect to η on each compact ).
(II) , 0 < (t − β) ≪ 1, , |q| ≤ p:

Since

()

then taking into account the limitation of the function Ψ(β, ζ; ξ, τ) with respect to the variables (β; ζ; ξ) on the set

with fixed t and τ, τ < t, we find that (82) will be fulfilled if the following relations are fulfilled:

()

The first boundary relation with (84) is obvious.

According to Lemma 8 statement, for all and , we have the following estimates:

()

Therefore, the second boundary relation from (84) also holds.

Thus, the fulfillment of statement (I) is established.

Further, according to inequality (46), for all , , τ ∈ [0, T), t ∈ (τ; T], and , we find the following estimate:

()

where the value of c_k,q > 0 does not depend on ξ, η, t, and τ. Hence, using the inequality

()

and emphasizing everywhere the dependence on ξ, η, τ, t, and β, we obtain that

()

for all

, τ ∈ [0, T), t ∈ (τ; T], and (t + τ)/2 ≤ β < t. Here the value of c_k,l > 0 does not depend on τ, t, β, and η.

Therefore, statement (II) is also fulfilled.

Lemma 10. The volume potential W is a differentiable function by the variable t on (τ; T] such that

()

Proof. Let us use the definition of the derivative:

()

It is necessary to prove the existence of this limit. However, the specified limit will exist if the corresponding one-sided limits I₊ and I₋ are equal to each other:

()

First, we check if there exists the boundary I₊. For this, we will use the following representation:

()

Hence, according to the statement of Lemma 9 and the average value of the integral, we arrive at the following ratios:

()

Thus, taking into account the continuity of the derivative by the variable t (see Lemma 9), as well as the equality (74), to prove the correctness of

()

it is enough to substantiate the limit transition in the following equality:

()

To do this, we estimate the integrand from the left side of this equality. Using the equality (74) and the fact that is a solution of equation (4) again, as well as estimates (73), (12), and (46), highlighting everywhere the dependence on ζ, we obtain the following estimate:

()

where t₁ = (t + τ)/2, ζ < (t − τ)/2, t ∈ (τ; T], τ ∈ [0, T) and

The obtained estimate characterizes the uniform convergence of the integral from the left side of the equality (95) and ensures the fulfillment of this equality.

Therefore, the right-hand limit I₊ exists, and

()

In a similar way, using the representation

()

we make sure of the existence of the left-hand limit I₋, the value of which coincides with I₊.

5. Green’s Function of the Cauchy Problem

We formulate the main result in the form of the following statement.

Theorem 11. The function Z defined by equality (16) is the GFCP for equation (2).

Proof. Taking into account that is a solution of equation (4) and taking into account equality (18), Lemma 10, and Corollary 7, for all , τ ∈ [0, T), and (x; t) ∈ Π_{(τ, 0]}, we obtain the correctness of such transformations:

()

Therefore, the function Z(x, t; ξ, τ), as a function of the variables (x; t) on the set Π_{(τ, T]}, is an ordinary solution of equation (2) at each fixed point (τ; ξ) ∈ Π_{(τ, T]}.

Now we make sure that the boundary relation

()

is satisfied (here the brackets 〈, 〉 indicate the result of the action of the functional on the basic function).

Since is the GFCP, then the following relation is correct:

()

Therefore, to prove statement (100), it is sufficient to establish the following boundary relation:

()

By analogy to establishing estimate (2.47) from [21], we arrive at the existence of a constant c₀ such that for all β ∈ [τ; t], 0 ≤ τ < t ≤ T, and , the estimate

()

is fulfilled. Here, the upper dash indicates complex conjugation.

Using the estimates (12), (103) and the corresponding Fubini theorem, we obtain the following estimates:

()

In this way, we come to the implementation of the ratio (102).

Finally, we note that the order p₁ of the group of junior members of the equations considered by Zhitomirskii in [3] when μ < 0 satisfies condition

()

and therefore the class of equations (2) considered here extends the class of Zhitomirskii parabolic Shilov-type equations with negative genus.

6. Conclusions

The class of differential equations (2) with dissipative parabolicity and negative genus considered in the paper is quite wide. It unifies and significantly expands the known classes of Shilov and Zhitomirskii parabolic equations with genus μ < 0. The parabolic stability of such equations with respect to the change of the junior coefficients and the estimate (12) of the parametric are sufficient to ensure the convergence of the process of successive approximations of the construction of the classical GFCP Z. Also, these conditions made it possible to find out the necessary smoothness of the function Z(x, t; ξ, τ) in a traditional way and to substantiate its property of δ-similarity with respect to t.

Estimates (51) of the derivatives of Z obtained here allow us to draw an important conclusion that even in the case of the negative genus μ, the GFCP Z has the typical exponential behavior for parabolic equations with respect to the spatial variable x.

The presence of GFCP Z allows us to successfully solve a number of important problems of the classical theory of the Cauchy problem for equation (2). In particular, to find sets of initial data of the Cauchy problem with which it will be correctly solved, to study the properties of the solutions of this problem, and to solve certain optimization problems for the equations (2).

Disclosure

This study was part of the employment at Chernivtsi National University.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Open Research

Data Availability

The data used in the research to support the findings of this study are purely bibliographic and from scientific publications, which are included in the article with their respective citations.

References

1 Shilov G. E., On conditions of correctness of Cauchy’s problem for systems of partial differential equations with constant coefficients, Uspekhi Matematicheskikh Nauk. (1955) 10, no. 4, 89–100.
Google Scholar
2 Gelfand I. and Shilov G., Generalized Functions, Theory of Differential Equations. (1967) 3, Academic Press, Boston, MA, USA.
Google Scholar
3 Zhitomirskii Y. I., Cauchy problem for some types of systems of linear equations, which are parabolic by G. E. Shilov, with partial derivatives with continuous coefficients, Izvestiya Rossiiskoi Akademii Nauk. (1959) 23, 925–932.
Google Scholar
4 Hou-sin U., On the definition of parabolicity of systems of equations with partial derivatives, Uspekhi Mat. Nauk. (1960) 15, no. 6, 157–161.
Google Scholar
5 Eidelman S. D., Ivasishen S. D., and Porper F. O., The Liouville theorems for systems parabolic by Shilov, Central European Journal of Mathematics. (1961) 169, no. 6.
Google Scholar
6 Gorodetskii V. V., Some theorems on stabilization of solutions of the Cauchy problem for Shilov parabolic systems in classes of generalized functions, Ukrainian Mathematical Journal. (1988) 40, no. 1, 43–48.
10.1007/BF01056443
Google Scholar
7 Gorodetskii V. V., The Cauchy problem for Shilov parabolic equations in classes of generalized periodic functions, Central European Journal of Mathematics. (1988) 10, no. 5, 82–84.
Google Scholar
8 Zheng Q., Matrices of operators and regularized cosine functions, Journal of Mathematical Analysis and Applications. (2006) 315, no. 1, 68–75.
10.1016/j.jmaa.2005.10.007
Google Scholar
9 Kostic M., Shilov Parabolic Systems, Bulletin. (2012) 37, 19–39.
Google Scholar
10 Dovzhytska I. M., The Cauchy problem for inhomogeneous parabolic Shilov equations, Carpathian Mathematical Publications. (2021) 13, no. 2, 475–484, https://doi.org/10.15330/cmp.13.2.475-484.
10.15330/cmp.13.2.475-484
Google Scholar
11 Unguryan G., Modified cauchy problem with impulse action for parabolic shilov equations, International Journal of Mathematics and Mathematical Sciences. (2021) 2021.
10.1155/2021/5539676
Google Scholar
12 Unguryan G. M., Nonlocal problem with impulse action for parabolic equations of vector order, Ukrains’kyi Matematychnyi Zhurnal. (2021) 73, no. 11, 1532–1540, https://doi.org/10.37863/umzh.v73i11.6521.
10.37863/umzh.v73i11.6521
Google Scholar
13 Petrovskii I. G., On the Cauchy problem for systems of equations with partial derivatives in a domain of nonanalytic functions, Bulletin. (1938) 1, no. 7, 1–72.
Google Scholar
14 Friedman A., Partial Differential Equations of Parabolic Type, 1964, Prentice-Hall, Englewood Cliffs, NJ, USA.
Google Scholar
15 Eidelman S. D., Parabolic Systems, 1969, North-Holland Publishing Company, Amsterdam, Netherland.
Google Scholar
16 Cho S., Dong H., and Kim S., On the green’s matrices of strongly parabolic systems of second order, Indiana University Mathematics Journal. (2008) 57, no. 4, 1633–1678.
10.1512/iumj.2008.57.3293
Google Scholar
17 Dong H., Escauriaza L., and Kim S., On C^1/2,1, C^1,2, C^0,0 estimates for linear parabolic operators, Journal of Evolution Equations. (2021) 21, no. 4, 4641–4702.
10.1007/s00028-021-00729-8
Google Scholar
18 Dong H., Kim S., and Lee S., Estimates for fundamental solutions of parabolic equations in non-divergence form, Journal of Differential Equations. (2022) 340, no. 340, 557–591.
10.1016/j.jde.2022.09.007
Google Scholar
19 Gelfand I. and Shilov G., Generalized functions, Spaces of Fundamental and Generalized Functions. (1968) 2, Academic Press, Boston, MA, USA.
Google Scholar
20 Litovchenko V. A., Peculiarities of the fundamental solution of parabolic systems with a negative genus: chapter of the monograph, Advances in the Solution of Nonlinear Differential Equations. (2021) IntelOpen, London, UK, https://doi.org/10.5772/intechopen.95024.
10.5772/intechopen.95024
Google Scholar
21 Litovchenko V. A. and Dovzhitska I. M., The fundamental matrix of solutions of the Cauchy problem for a class of parabolic systems of the Shilov type with variable coefficients, Journal of Mathematical Sciences. (2011) 175, no. 4, 450–476, https://doi.org/10.1007/s10958-011-0356-0, 2-s2.0-79957657646.
10.1007/s10958-011-0356-0
Google Scholar

All articles

Green’s Function of the Cauchy Problem for Equations with Dissipative Parabolicity, Negative Genus, and Variable Coefficients

Abstract

1. Introduction

2. Preliminary Information

3. Method of Successive Approximations

4. Properties of the Volume Potential in relation to the Time Variable

5. Green’s Function of the Cauchy Problem

6. Conclusions

Disclosure

Conflicts of Interest

Open Research

Data Availability

References

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

Green’s Function of the Cauchy Problem for Equations with Dissipative Parabolicity, Negative Genus, and Variable Coefficients

Abstract

1. Introduction

2. Preliminary Information

3. Method of Successive Approximations

4. Properties of the Volume Potential in relation to the Time Variable

5. Green’s Function of the Cauchy Problem

6. Conclusions

Disclosure

Conflicts of Interest

Open Research

Data Availability

References

References

Related

Information