Chapter 1 Preliminary
This chapter will introduce some basic results， most of which will be used in the following chapters. First we shall recall some basic inequalities whose detailed proofs can be found in the related literature， see， e.g.， Adams [1，2]， Friedman [37， 38]， Gagliardo [40，41]， Nirenberg [95， 96]， Yosida [148]， etc.
1.1 Some Basic Inequalities
1.1.1 The Sobolev Inequalities
We shall first introduce some basic concepts of Sobolev spaces.
Definition 1.1.1. Assume is a bounded or an unbounded domain with a smooth boundary r. For 1 < p < +co and m a non-negative integer， is
defined to be the space of functions u in LP(Q) whose distribution derivatives of order up to m are also in LP(Q). That is.
The space called a Sobolev space， is equipped with a norm
(1.1.1)
(1.1.2)
which is clearly equivalent to
(1.1.3)
If， we only denote is a Banach space. The space Wom，is defined as the closure of relative to the norm (1.1.3). Clearly，with norm， is a Hilbert space with respect to the scalar product with， here g is the conjugate function of g.
It is well-known that the Sobolev inequalities are important tools in the study of nonlinear evolutionary equations. First， we shall introduce these inequalities for functions in the space
Theorem 1.1.1 (The Sobolev Inequality). Assume that，is an open domain. There exists a constant such that
(1)
(1.1.4)
where;
(2) if p>n and Q is bounded， and u G then u G C(Q) and
(1.1.5)
While， then
(1.1.6)
where measure the n-dimensional unit ball， T is the Euler gamma function and.
Remark 1.1.1. The Sobolev inequality (1.1.4) does not hold for p = n， p* = +oo.
(1.1.4) was first proved by Sobolev [138] in 1938. Sobolev [138] stated that the Lp* norm of u can be estimated by， the Sobolev norm of u. However， we can bound a higher Lp norm of u by exploiting higher order derivatives of u as shown in the next theorem which generalizes theorem 1.1.1 from m = 1， p > n to an integer.
Theorem 1.1.2. Assume QCMn is an open domain. There exists a constant C = C(n，p) > 0 such that
(1) if， and u G then and
(1.1.7)
(2) if mp > n， and u S VF0m，p(n)， then and
(1.1.8)
where and diamK is the diameter of K.
(1.1.8)
Remark 1.1.2. An important case considered in theorems 1.1.1 and 1.1.2 is Q = Mw. In this situation， and therefore the results of theorems 1.1.1 and 1.1.2 apply to.
For p > n， the results of theorems 1.1.1 and 1.1.2 imply the fact that u is bounded. Indeed， u is Hoder continuous， which we shall state as follows.
Theorem 1.1.3. If ue， then where.
Generally， the embedding theorems are closely related to the smoothness of the domain considered， which means that when we study the embedding theorems， we need some smoothness conditions for the domain. These conditions include that the domain Q possesses the cone property， and it is a uniformly regular open set in Mra， etc. For example， when or Lip， Q has the cone property. Mathematically， we need to define the special meaning of the word “embedding” or “compact embedding”.
Definition 1.1.2. Assume A and B are two subsets of some function space. Set A is said to be embedded into B if and only if
(1)A C B;
(2) the identity mapping I: A B is continuous，i.e.， there exists a constant C > 0 such that for any x ^ A， there holds that.
If A is embedded into B， then we simply denote by.
A is said to be compactly embedded into B if and only if
(1) A is embedded into B;
(2) the identity mapping I: is a compact operator.
If A is compactly embedded into B，then we simply denote by A B.
Now we draw some consequences from theorem 1.1.1. In fact， exploiting theorem 1.1.1， we have the following result which is an embedding theorem.
Corollary 1.1.1. If then u G Lq(Q) with， and. Moreover，if p > n，u coincides. in Q with a (uniquely determined) function of C(Q). Finally，there holds that，
(1.1.9)
(1.1.10)
(1.1.11)
where C = C(n，p，q) > 0 is a constant.
We can generalize corollary 1.1.1 to functions from VF0m，p(O) which can be stated as the following embedding theorem.
Theorem 1.1.4. Let. Then
(1)
(1.1.12)
and there is a constant C± > 0 depending only on m，p，q and n such that for all
P，
(1.1.13)
(2) if mp = n， then we have，for all，
(1.1.14)
and there is a constant C2 > 0 depending only on m， p， q and n such that for all，
(1.1.15)
(3) if，each is equal a.e. in D. to a unique function in Ck(Q)， for all and there is a constant C3 > 0 depending only on m， p， q and n such that
(1.1.16)
Remark 1.1.3. In case (2) of theorem 1.1.4， the following exception case holds for
(1.1.17)
Now we give the following compact embedding theorem.
Theorem 1.1.5 (Embedding and Compact Embedding Theorem). Assume that Q is a bounded domain
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