In this section, we obtain the well-posedness and large time behavior of solutions to (1.1) for , and we prove Theorems 1.1-1.2. To do this, we first obtain the well-posedness to (1.1) by the following proposition.
Proposition 3.1 Assume that , , , and satisfies (1.2). Then there exists a unique solution to (1.1) in the sense of Definition 1.1.
Proof The idea of the proof is same as in [25, 28], so we omit here. □
Next, in order to prove the large time behavior of the solution to (1.1), we shall need several steps: Step 1, we will find that the temporal decay rate of is indeed . Step 2, we prove the boundary estimates for the large time which guarantee that no loss of boundary condition occurs throughout the time evolution. Step 3, the half-relaxed limits technique is applied to show the expected convergence after introducing self-similar variables. The approach is developed by Laurençot and Stinner in [25, 27]. To do this, we need the following lemmas.
Lemma 3.1 (Upper bound)
Assume that , , , and the initial data satisfies (1.2). Then there exists depending only on m, l, q, Ω, and such that
(3.1)
Proof Assume that and such that . For , we define the function
where and satisfies the following condition:
(3.2)
Since , the function is -smooth in . Moreover, according the condition , , and , we have and . Therefore, for and , it follows from (3.2) that
(3.3)
Hence, the condition (3.2) guarantees that is a supersolution to (1.1) in . In addition, since for , for , we deduce from (3.2) that
and
By the comparison principle, we have
The proof of Lemma 3.1 is complete. □
Lemma 3.2 (Upper bound)
Assume that , , , and the initial data satisfies (1.2). Then there exists depending only on m, l, q, Ω, and such that
(3.4)
Proof Assume that and such that . For , we define the function
where the positive constants , R, and δ satisfy the following condition:
(3.5)
Since , the function is -smooth in . Moreover, for and , it follows from (3.5) and that
(3.6)
Therefore, the condition (3.5) guarantees that is a supersolution to (1.1) in . In addition, since for , for , we deduce from (3.5) that
and
By the comparison principle, we have
The proof of Lemma 3.2 is complete. □
Lemma 3.3 (Hölder estimate)
Assume that , , , and the initial data satisfies (1.2). Then there exists depending only on m, l, q, Ω, and such that
(3.7)
Proof Since the boundary ∂ Ω of Ω is smooth, there exists such that for each , there exists satisfying and . It follows from the initial data condition that is Lipschitz continuous, i.e., there exists such that
(3.8)
Next, we define the open subset of by
where satisfies , and denote the function
for and .
Moreover, we assume that
(3.9)
and
(3.10)
where , , and are defined in Lemmas 3.1 and 3.2, respectively.
Since , the function is -smooth in . For , we have . By a direct computation, we infer from (3.9)-(3.10) that
Therefore, is a supersolution to (1.1) in . In addition, it follows from (3.8)-(3.10) and the mean value theorem that
Moreover, for , we have either or . If , then we have
If , by Lemmas 3.1-3.2 and (3.9), we have
By the comparison principle [6] we have
Consequently,
for .
Since , we have
(3.11)
Finally, we consider and . If , it follows from Lemmas 3.1-3.2 that
where , and and are defined in Lemmas 3.1 and 3.2, respectively.
If , let satisfy and . Since , we have
which implies . Therefore, we deduce from (3.11) that
Choosing , we have
The proof of Lemma 3.3 is complete. □
We next proceed as in [29] to deduce the Hölder continuity of from Lemma 3.3. Therefore, we obtain the following corollary.
Corollary 3.1 Assume that , , , and the initial data satisfies (1.2). Then there exists depending only on m, l, q, Ω, and such that
(3.12)
Proof The proof is similar to the argument in [25, 29], so we omit here. □
Proofs of Theorems 1.1 and 1.2 The proofs are based on the ideas in [25], but we give the details of the argument for the reader’s convenience. Let be the solution to the porous medium equation with homogeneous Dirichlet boundary conditions
(3.13)
According to the nonnegativity of , it follows from the comparison principle [6] that
(3.14)
We introduce the scaling variable for and denote the new unknown function v and V by
and
Then v is a viscosity solution to the following problem:
(3.15)
In addition, owing to Lemmas 3.1-3.3 and (3.14), we have
(3.16)
and
(3.17)
Next, for , we define
and the half-relaxed limits
for .
By (3.16), it is easy to see that and are well-defined and do not depend on . Moreover, it readily follows from (3.15) and (3.17) that
(3.18)
By a direct computation, is a solution to the following initial-boundary problem:
(3.19)
Next, we shall give proofs of Theorems 1.1 and 1.2. To do this, we distinguish the two cases and .
Case 1. . We use the stability of semicontinuous viscosity solutions [6] to deduce from (3.19) that
(3.20)
and
(3.21)
In addition, by [30], in as . Moreover, it follows from (3.16) and the definition of and that
(3.22)
Since in Ω by [30], we deduce from (3.22) that and are positive and bounded in Ω and vanish on ∂ Ω by (3.18). Owing to (3.20) and (3.21), we infer from Lemma 2.1 that
By (3.22), we have
Setting , we deduce from (3.18), (3.20), (3.21), and (3.22) that is a positive viscosity solution to (2.1) and solves (1.9). Therefore, the existence of a positive solution to (1.9) is proved. Moreover, by Corollary 2.1, we obtain the uniqueness of solution to (1.9).
Furthermore, it follows from the equality that
i.e.,
(3.23)
Therefore, we infer from the scaling transformation that
(3.24)
Finally, Corollary 3.1 gives the last statement of Theorem 1.1. The proof of Theorem 1.1 is complete.
Case 2. . We use once more the stability of semicontinuous viscosity solutions [6] to deduce from (3.19) that
(3.25)
and
(3.26)
In addition, by [30], in as . Moreover, it follows from (3.16) and the definition of and that
(3.27)
Since in Ω by [30] and is a solution to (2.10), we can apply Lemma 2.2 to conclude that in . Owing to (3.27), we have
Therefore, we deduce from (3.19) that
i.e.,
(3.28)
Thus, we infer from the scaling transformation that
(3.29)
The proof of Theorem 1.2 is complete. □