Self-Adjointness, Symmetries, and Conservation Laws for a Class of Wave Equations Incorporating Dissipation

Table 1

Nonlinear self-adjointness of (3).

𝑓	𝑔	𝑣	Self-adjointness
∀(𝑓𝑢≠0)	∀	𝐶1+𝐶2𝑒^𝛼𝑡	Nonlinear
∀(𝑓𝑢≠0)	𝐶5	(𝐶1+𝐶2𝑒^𝛼𝑡)(𝐶3+𝐶4𝑥)	Nonlinear
∀(𝑓𝑢≠0)	𝐶5∫𝑓(𝑢)𝑑𝑢	𝐶1+𝐶2𝑒^𝛼𝑡+(𝐶3+𝐶4𝑒^𝛼𝑡)𝑒^𝑥/𝐶5	Nonlinear
𝐶0(≠0)	𝐶1+𝐶2𝑢	𝐶1𝑒^(𝐶2/𝐶0)𝑥+𝛼𝑡𝑢+𝑏(𝑡,𝑥)	Weak

𝛼 is a nonzero constant; 𝑏(𝑡,𝑥) satisfies (27).