In this paper, we consider the three-dimensional Cauchy problem of the nonisentropic compressible Euler equations with relaxation. Following the method of Wu et al. (2021, Adv. Math. Phys. Art. ID 5512285, pp. 1–13), we show the existence and uniqueness of the global small
H
k
k
⩾
3
solution only under the condition of smallness of the
H
3
norm of the initial data. Moreover, we use a pure energy method with a time-weighted argument to prove the optimal
L
p
–
L
q
1
⩽
p
⩽
2
,
2
⩽
q
⩽
∞
-type decay rates of the solution and its higher-order derivatives.
We study the Cauchy problem of the three-dimensional full compressible Euler equations with damping and heat conduction. We prove the existence and uniqueness of the global small
H
N
N
≥
3
solution; in particular, we only require that the
H
4
norms of the initial data be small when
N
≥
5
. Moreover, we use a pure energy method to show that the global solution converges to the constant equilibrium state with an optimal algebraic decay rate as time goes to infinity.
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