For Cauchy problem of multi-dimensional scalar conservation laws, Conway and Smoller [4] and Kruzkov [1] have proved that weak solution uniquely exists if it also satisfies entropy condition, and it is called weak entropy solutions.
It is well known that if T is bounded and has closed range, then (T^{dagger }) uniquely exists and it is continuous.
Furthermore, they obtained the basic reproduction number (Re_{0}), which played as a threshold parameter and satisfied that if (Re_{0}<1), then the disease-free equilibrium of system (1.1) is locally and globally asymptotically stable, while if (Re _{0}>1), then the endemic equilibrium uniquely exists and it is locally and globally asymptotically stable.
We show the time-local well-posedness for a system of nonlinear dispersive equations for the water wave interaction[formula]It is shown that for any initial data (u0, v0)∈Hs(R)×Hs−1/2(R) (s⩾0), the solution for the above equation uniquely exists in a subset ofC((−T, T);Hs)×C((−T, T);Hs−1/2) and depends continuously on the data.
The solution of (1.6) uniquely exists.
τ* ∈ (0,T) uniquely exists under Assumption 3.2 and (11).
That is, there uniquely exists trivial solution satisfying (3.8).
Hence, we know that (v_{1}) uniquely exists and is positive.
Then there uniquely exists p ∈ Ω such that Q p 2 x = s [25, 26].
Therefore, solution ((S(n),E(n),I(n),R(n))) uniquely exists and is positive for all (n>0).
