A:  From The Urantia Book:

The life bestowed upon plants and animals by the Life Carriers does not return to the Life Carriers upon the death of plant or animal. The departing life of such a living thing possesses neither identity nor personality ; it does not individually survive death. During its existence and the time of its sojourn in the body of matter, it has undergone a change; it has undergone energy evolution and survives only as a part of the cosmic forces of the universe; it does not survive as individual life. The survival of mortal creatures is wholly predicated on the evolvement of an immortal soul within the mortal mind. (36:6.5)

That afternoon Jesus and Ganid had both enjoyed playing with a very intelligent shepherd dog, and Ganid wanted to know whether the dog had a soul , whether it had a will, and in response to his questions Jesus said: “The dog has a mind which can know material man, his master, but cannot know God, who is spirit; therefore the dog does not possess a spiritual nature and cannot enjoy a spiritual experience. The dog may have a will derived from nature and augmented by training, but such a power of mind is not a spiritual force, neither is it comparable to the human will, inasmuch as it is not reflective–it is not the result of discriminating higher and moral meanings or choosing spiritual and eternal values. It is the possession of such powers of spiritual discrimination and truth choosing that makes mortal man a moral being, a creature endowed with the attributes of spiritual responsibility and the potential of eternal survival.” Jesus went on to explain that it is the absence of such mental powers in the animal which makes it forever impossible for the animal world to develop language in time or to experience anything equivalent to personality survival in eternity. As a result of this day’s instruction Ganid never again entertained belief in the transmigration of the souls of men into the bodies of animals. (130:2.8)

As much as we might hope that our pets and other animals will be with us on the eternal journey, it appears that this is not the case. However, there will likely be other fascinating creatures that we will encounter in our eternal life that may be somewhat akin to our pets. And, I feel sure that the memories of the love and companionship that we enjoyed with our beloved pets on earth will endure forever.”