#### Abstract

Conditional Lie-Bäcklund symmetry approach is used to study the invariant subspace of the nonlinear diffusion equations with source , . We obtain a complete list of canonical forms for such equations admit multidimensional invariant subspaces determined by higher order conditional Lie-Bäcklund symmetries. The resulting equations are either solved exactly or reduced to some finite-dimensional dynamic systems.

#### 1. Introduction

The classical symmetry theory for studying differential equations is presented firstly by Lie, which has been universally used and proved to be very effective in similarity reductions and group classifications [1–7]. However, there exist some important equations with very small Lie point symmetry groups. For example, the Fisher equation and Fitzhugh-Nagumo equation, which are widely used in mathematical biology, are invariant only under the time and space translations. This means that the classical symmetry reduction method is not a proper tool for dealing with these equations. To overcome this difficulty, several generalized methods have been developed and established, including the nonclassical symmetry method (or referred to as the conditional symmetry method) [8], the weak symmetry method [9, 10], iteration of the nonclassical method [11], the Clarkson-Kruskal direct method [12, 13], and the conditional Lie-Bäklund symmetry (CLBS) method (or referred to as the generalized conditional symmetry method) [14–16].

CLBS can be regarded as a natural generalization of the nonclassical symmetry. Therefore, the procedure for computing the CLBSs is about the same as for the nonclassical method. Furthermore, Galaktionov and Svirshchevski have shown that the CLBS method is closely related to the invariant subspace (IS) method; namely, exact solutions defined on ISs for differential equations or their variant forms can be obtained by using the CLBS method [17–19]. For nonlinear diffusion equations (NLDEs), symmetry-related methods, especially the CLBS method, have been proved to be very powerful to classify and reduce the considered equations [20–34]. For example, NLDEs can be used to describe not only the process by which matter is transported from one part of a system to another, as a result of random molecular motion, but they can also represent many other apparently unrelated phenomena such as heat conduction in solids or even the stationary notion of a boundary layer of fluid over a flat plate [35]. In [36], the Lie point symmetry method has been used to obtain the similarity solutions of the inhomogeneous NLDEs belonging to the above equations, where nonzero constants and have several applications such as propagation of a thermal wave in an exponential atmosphere. A complete classification of the symmetry reductions of these equations using the nonclassical method is given by Saied in [37]. The second-order CLBSs of these equations have also been studied in [33]. Furthermore, the generalized porous medium equations are considered by using the CLBS method in [34]. Some exact solutions, defined on the polynomial, trigonometric, and exponential ISs determined by the CLBSs, are constructed.

In this paper, we mainly discuss the following NLDEs: by means of the CLBS method. Here, and are, respectively, referred to as the diffusion and source terms. Equation (4) has a wide range of applications in physics, diffusion process, and engineering sciences and has been applied to describe several situations such as heat conduction by electrons in a plasma, heat conduction by radiation in a fully ionized gas, axisymmetric flow of a very viscous fluid, and turbulent diffusion [38, 39].

The remainder of this paper is organized as follows. In the following section, we review some necessary notations, definitions, and fundamental theorems on the CLBS method. Equations of the form (4) admitting CLBSs and the corresponding ISs are classified in Section 3. Exact solutions and reductions of some examples in the resulting equations are obtained in Section 4. The last section is devoted to conclusions and discussions.

#### 2. Preliminaries

Let us give a brief discussion on the CLBS method. For the th-order equation we set as an evolutionary vector field with characteristic . Here, we use the following notations:

*Definition 1. *The evolutionary vector field (6) is said to be a Lie-Bäcklund symmetry of (5) if and only if
where is the set of all differential consequences of the equation; that is,

*Definition 2. *The evolutionary vector field (6) is said to be a CLBS of (5) if and only if
where denotes the set of all differential consequences of equation with respect to ; that is, , .

Proposition 3 (Zhdanov [14] and Fokas and Liu [15, 16]). *Equation (5) admits the CLBS (6) if there exists a function such that
**
where , the prime denotes the Fréchet derivative, and is an analytic function of , , , , and , , .*

An obvious conclusion of this proposition is that (5) admits the CLBS with the characteristic if Here, and are given as in Definitions 1 and 2.

For (4), we set the characteristic where , , and . It is important to note that if (4) admits CLBS (13), then equation admits CLBS In fact, (4) and (14) are related as where denotes the inverse function of .

From (12), we can see that (14) admitting of the CLBS with the characteristic (15) is equivalent to ; namely, where and is given as in Definition 2. Thus, the linear solution space of linear ordinary differential equation (ODE) is invariant with respect to the above operator ; that is, It follows that if (14) admits CLBS (15), then (14) has an exact solution of the generalized separation of variables form where the coefficients satisfy the -dimensional dynamic system

Thus, we will determine the forms of (14) so that (14) admits the CLBS (15), which is equivalent to classify (14) in terms of the IS (19), or it is equivalent to study the CLBS (13) of (4).

The following theorem provides us with the estimate to the maximal dimension of the IS admitted by an th-order nonlinear differential operator.

Theorem 4 (Galaktionov and Svirshchevski [19]). *If a linear space is invariant with respect to the nonlinear differential operator of order , then there exists an inequality
*

It follows that the order of linear ODE is not greater than five if (14) admits the CLBS (15). This allows us to classify (14) based on the existence of the generalized variable separation solutions (GVSSs) (21), which are generated by the solution space (19) determined by the linear ODE , or it is equivalent to study the GVSSs of (4) generated by .

#### 3. CLBSs and Corresponding ISs of (14)

In view of Theorem 4, it suffices to consider CLBSs (15) of (14) with . We first consider the case . It implies from (17) that (14) admits the CLBS (15) if there holds where the primes and subscripts denote the derivatives and the partial derivatives with respect to the indicated variables, respectively. To vanish all the coefficients of (24), we will have the following overdetermined system: For general , it is apparent that and . Substituting into the second of the above system, we arrive at To solve (26), we can derive three possibilities: , , and .

Assume that the diffusion coefficient takes power or exponential forms. From (16), without loss of generality, it is reasonable to consider the following four cases: (i) , ; (ii) , ; (iii) , ; and (iv) , , where and are arbitrary constants.

*Case 1 (). *In this case, we can derive . Correspondingly, there exist the following cases: (a) , ; (b) , ; and (c) , . If , , the third and fourth equations are simplified as
Then, we derive or . However, if , function turns to zero, which should be omitted. By the similar calculation, we obtain or with case (b), while we derive with case (c). Therefore, we have results listed as the 1–6th entries in Table 1 with Case 1.

*Case 2 (). *In this case, (26) becomes , which implies or . When , we have ; the corresponding solution is listed as the seventh entry in Table 1 with arbitrary . And results in the case of are presented as the 8–11th entries in Table 1.

*Case 3 (). *In this case, we can derive
which arrives at or , . When , it is easy to solve out . Without loss of generality, we consider two cases (a) , , and (b) , . If , , we can formulate according to the third equation of (25). Then, substituting into the fourth equation, we have the following condition:
Therefore, for general , we have . So, we derive the corresponding result as the 12-13th entries in the Table 1. By similar calculation, we obtain solution listed as 14-15th entries in Table 1 with the case of , . When , , it is apparent that ; we assume , without loss of generality. Then, we substitute and into the third equation of (25), acquiring . After translation transformation of , we have , which can be seen as the same case with the former one. Substituting into the fourth equation, we find it identical spontaneously. The corresponding result is listed as the 16th entry in Table 1.

Finally, we have all the solutions listed in the Table 1 with case of for general including the special relation between and , where the unknown functions are given as , for , and , for ; , for , and , for ; , for and , for .

For special , it is noted that we can combine some terms in (24). This leads to new overdetermined system which is different from (25). Therefore, there exist new forms of (14) admitting CLBS (15) and IS (19) with , which are not included in Table 1.

For , it follows from (24) that the coefficients in (14) and (15) satisfy Solving the above system by the same approach, we can obtain new results which are presented in Table 2. Similarly, we also list new results in Table 2 for other special .

The unknown functions in Table 2 are given in the following:

When , by similar calculation, we can derive the classification result listed in Table 3, where , for , and , for ; , , for , and , , for , and , , for ; , , , for , and , , , for .

For , we find that the overdetermining system is inconsistent. So, there are no CLBSs (15) and ISs (19) for (14).

#### 4. CLBS (13) and Reductions of (4)

For (16), we can transfer the CLBS (15) of (14) into CLBS (13) of (4); that is, where is the inverse function of as referred above. Hence, the GVSS of (4) can be derived from the GVSS (21) with the transformation . Here, the GVSS (21) is defined on IS (19) determined by the linear ODE , where is given by CLBS (15). The coefficient functions are determined by a finite-dimensional dynamic system. For simplification, we just give some examples to illustrate our approach. Here, we pointed out that the selection of examples is random.

*Example 1. *Equation
admits the CLBS

The corresponding GVSS is given by where and satisfy the 2-dimensional dynamic system When , we have with arbitrary constants and . If , we have with arbitrary constants and .

*Example 2. *Equation
with admits the CLBS

The corresponding GVSS is given by where and satisfy the 2-dimensional dynamic system

*Example 3. *Equation
with admits the CLBS

The corresponding GVSS is given by where and satisfy the 2-dimensional dynamic system

*Example 4. *Equation
admits the CLBS

The corresponding GVSS is given by where , , and satisfy the 3-dimensional dynamic system This system has the following exact solution:

*Example 5. *Equation
admits the CLBS

The corresponding GVSS is given by where , , and satisfy the 3-dimensional dynamic system

*Example 6. *Equation
admits the CLBS

The corresponding GVSS is given by where , , and satisfy the 3-dimensional dynamic system

*Example 7. *Equation
admits the CLBS

The corresponding GVSS is given by where , , , and satisfy the following 4-dimensional dynamic system:

#### 5. Conclusions and Discussions

In this paper, we have applied the CLBS method with ISs to study (4). The transformed equations (14) admitting CLBSs (15) are listed in Tables 1, 2, and 3. The corresponding reduced equations of the resulting equations are finite-dimensional dynamic systems defined on . Some concrete examples are illustrated in Section 4. Generally speaking, these reductions cannot be obtained in the frameworks within the Lie point symmetry method and the nonclassical symmetry method. Of course, we mention that the asymptotical behavior, blow-up, extinction, and geometric properties for these finite-dimensional dynamic systems are worthy of further study.

For NLDEs, we find that the CLBS method plays a key role in the study of their asymptotical behavior, blow-up, extinction, and geometric properties because of the diversity of solutions obtained by this method. Although this is an effective and complete method, there are still some important problems to be studied. How to study higher-dimensional NLDEs and systems via the CLBS method? How to deal with the initial value problems by means of the CLBS method? Is it possible to apply the CLBS method to other types of evolution equations?

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant no. 11371323) and the Natural Science Foundation of Zhejiang province (Grant no. Y6100611, Q12A010084).