Solvability of the Cauchy Problem for a Quasilinear System in Original Coordinates
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Journal of Mathematical Sciences, Vol. 249, No. 6, September, 2020
SOLVABILITY OF THE CAUCHY PROBLEM FOR A QUASILINEAR SYSTEM IN ORIGINAL COORDINATES M. V. Dontsova Lobachevsky State University of Nizhny Novgorod 23, Gagarina Pr., Nizhny Novgorod 603950m Russia [email protected]
UDC 517.9
We study the Cauchy problem for a system of quasilinear equations in the original coordinates by using the additional argument method. We obtain sufficient conditions for the existence and uniqueness of a local solution and show that the solution has the same x-smoothness as the initial function. We also obtain sufficient conditions for the existence and uniqueness of a global solution. Bibliography: 4 titles.
1
Introduction
We consider the system ∂t u(t, x) + (a1 (t)u(t, x) + b1 (t)v(t, x))∂x u(t, x) = a2 u(t, x) + b2 (t)v(t, x),
(1.1)
∂t v(t, x) + (c1 (t)u(t, x) + g1 (t)v(t, x))∂x v(t, x) = g2 v(t, x), where u(t, x) and v(t, x) are unknown functions, a1 (t), b1 (t), b2 (t), c1 (t), g1 (t) are known functions, a2 and g2 are known constants. For the system (1.1) we consider the initial conditions u(0, x) = ϕ1 (x),
v(0, x) = ϕ2 (x),
(1.2)
where ϕ1 (x) and ϕ2 (x) are known. The problem (1.1), (1.2) is considered in the domain ΩT = {(t, x) | 0 t T, x ∈ (−∞, +∞), T > 0}. A similar problem was studied in [1]. In this paper, we get other sufficient conditions in the case of negative a1 (t), b1 (t), c1 (t), g1 (t) and nonnegative b2 (t) on [0, T ]. Using the additional argument method, we obtain a system of integral equations which is equivalent to the system considered in [1], but allowing one to prove estimates in a simper way. By the additional argument method, we consider the extended characteristic system dη1 (s, t, x) = a1 (s)w1 (s, t, x) + b1 (s)w3 (s, t, x), ds
(1.3)
dη2 (s, t, x) = c1 (s)w4 (s, t, x) + g1 (s)w2 (s, t, x), ds
(1.4)
Translated from Problemy Matematicheskogo Analiza 103, 2020, pp. 91-100. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2496-0918
918
dw1 (s, t, x) = a2 w1 (s, t, x) + b2 (s)w3 (s, t, x), ds
(1.5)
dw2 (s, t, x) = g2 w2 (s, t, x), ds
(1.6)
w3 (s, t, x) = w2 (s, s, η1 ),
(1.7)
η1 (t, t, x) = x,
w4 (s, t, x) = w1 (s, s, η2 ),
η2 (t, t, x) = x,
w1 (0, t, x) = ϕ1 (η1 (0, t, x)),
(1.8)
w2 (0, t, x) = ϕ2 (η2 (0, t, x)).
(1.9)
The unknowns ηi , wj , i = 1, 2, j = 1, . . . , 4, depend not only on t and x, but also on the additional variable s. Integrating (1.3)–(1.6) with respect to s and taking into account the conditions (1.7)–(1.9), we obtain the equivalent system of integral equations t η1 (s, t, x) = x −
(a1 (τ )w1 + b1 (τ )w3 )dτ,
(1.10)
(c1 (τ )w4 + g1 (τ )w2 )dτ,
(1.11)
s
t η2 (s, t, x) = x − s
s b2 (τ )w3 exp(a2 (s − τ ))dτ,
w1 (s, t, x) = ϕ1 (η1 (0, t, x)) exp(a2 s) +
(1.12)
0
w2 (s, t, x) = ϕ2 (η2 (0, t, x)) exp(g2 s),
(1.13)
w3 (s, t, x) = w2 (s, s, η1 ),
(1.14)
w4 (s, t, x) = w1 (s, s, η2 ).
(1.15)
Substituting (1.10), (1.11) into (1.12)–(1.15), we get
t
w1 (s, t, x) = ϕ1 x −
(a1 (τ )w1 + b1 (τ )w3 )dτ
exp(a2 s)
0
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