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In this paper a variable-coefficient reaction-diffusion equation is studied. We classify the equation into three kinds by different restraints imposed on the variable coefficient

Conservation law is an important concept in physics. It describes a quantity that is conserved; that is, the total amount is the same before and after something occurs. In mathematics, conservation law provides one of the basic principles in formulating and investigating models. For instance, sometimes, the existence of a large amount of conservation laws of a partial differential equation (PDE) is a strong indication of its integrability.

For an Euler-Lagrange equation, which arises out of a variational principle, the remarkable Noether’s theorem [

Among these generalizations, after suggesting some concepts, such as adjoint equation, strict self-adjointness, quasi-self-adjointness, and nonlinear self-adjointness, Ibragimov [

Following Ibragimov, many researchers have been studying this interesting area and there are a lot of works in the literature, such as [

In this paper, we consider a variable-coefficient reaction-diffusion equation [

In particular,

In [

We start our work with solving the system of determining equations to obtain the infinitesimal generator and meantime classifying (

The contents of this paper are as follows. In Section

We assume that all functions are smooth, and the summation over the repeated indices is understood.

In this subsection, we recall Ibragimov’s procedure of constructing conservation laws corresponding to the given symmetries of any system of PDEs, provided that the number of equations in the system is equal to the number of dependent variables.

For convenience, we consider a scalar evolution equation

The adjoint equation to (

We now extend (

Any Lie point, Lie-Bäcklund, or nonlocal symmetry

According to Theorem

Sometimes the multiplier

Equation (

Equation (

Equation (

Equation (

The first three definitions were suggested by Ibragimov in [

The theorem on the order of local conservation laws for a more general class of second-order evolution equations, which covers (

Any local conservation law of any second-order

In this section, we present the most general Lie group of point transformations, which leaves (

First of all, we consider a one-parameter Lie group of infinitesimal transformation:

Applying the second prolongation

Substituting

The solution of the Subsystem (

By substituting

According to

When

When

When

Now we study the self-adjoint property of (

The adjoint equation to (

Setting

Setting

Hence, for

We apply Theorem

For Case

For Case

For Case

According to the definition of conservation law, we say that we have obtained local conservation laws for the simultaneous system of (

In view of Theorem

In this section, we will construct scale-invariant solution for (

For (

If we get a solution of this equation, we can construct a scale-invariant solution of (

When we used the odeadvisor command of Maple for classifying (

Noting that (

The solutions (

If

This is clearly invariant that under the group of translations in the

Some exact solutions and a discussion on local conservation laws for the variable-coefficient reaction-diffusion equation were presented. Firstly, by solving the determining equations in Lie symmetry analysis, we classified the equation into three kinds. For every kind, we obtained the Lie symmetries. Corresponding to these Lie point symmetries, we have expressed the conservation laws, respectively. For (

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

The authors are grateful to the referee for his/her valuable comments and suggestions which have led to an improvement of the presentation. This research is supported by the National Natural Science Foundation of China (no. 11061016).