After the above preparation, we are in a position to prove our main theorem.
Proof To prove the theorem, we split the process into several steps. The first step is to construct the approximating solution sequence; in the second step, we prove the regularity of the solution we have obtained; and in the last step, we prove the uniqueness. To get the existence of a weak solution in the space , it is natural to require to be a Cauchy sequence in the strong topology of and to be bounded in , here . To estimate the norm of , we will use the Hardy inequality.
Step 1: We construct an iterative solution sequence to approximate the solution of the original equation.
Firstly, on the one hand, to guarantee the weak convergence, we need the weak convergence of in , which in turn can be deduced by the boundedness of in . By the Hardy inequality [10–12], we have
On the other hand, by Lemma 2.1 and Lemma 2.3, let , we obtain a unique solution to (3.1). To pass to the limit in the system of (1.1), we need a strong convergence. In order to get this, we compute the Cauchy sequence as follows:
Combining (3.2) with (3.3), we solve the indices and . We claim that
uniformly in n. Thus,
By a process of induction, we obtain
which implies converges to some f in .
Proof of the claim of (3.4).
A direct calculation by using the Hardy inequality yields
here , .
Differentiating equation (3.1) by and , respectively, we get
Applying Lemma 2.3 to (3.6) yields
where . Since
We require , , this is an easy thing. Note that and . Plugging (3.8) and (3.9) into (3.7) yields
Next, we only have to solve the Gronwall inequalities in the form
These inequalities only hold in finite time, we denote the maximal existence time by T, i.e., for , there exists such that
Therefore, the claim of (3.4) holds.
According to the standard weak convergence process, we conclude that f is a solution of the Cauchy problem of equations of (1.1).
Step 2: Regularity of the solution.
Denote or by D. Since in , which deduces in , note that , we have .
By property (i) of Proposition A.3 in , we conclude that is nonnegative. Moreover, , a.e. , since in , which implies that f is nonnegative.
Step 3: Uniqueness of the solution.
The uniqueness is a direct consequence of
which in turn is a result of a very similar process to (3.5).
Next we are going to deal with the boundary, i.e., to show that the solution satisfied the boundary condition.
On the one hand, by Lemma 2.3, we have for x a.e.; on the other hand, for x a.e. Thus for x a.e., note the assumption of in the sense of trace of , we conclude in the sense of trace of , since the boundary is and :