The generalized nonlinear heat equation and its spectrum
In this paper, we study the nonlinear equation of the form ∂/∂t u(s, t) + c2(-*)ku(x, t) = f(x, t, u(x, t)), where*k is the operator iterated k-times, defined by*k=[(Σi=1p ∂2/ ∂xi2)3 + (Σj=p+1 ∂2/∂xi2)3]k, where p + q = n is the dimension of the Euclidean space ℝn, u(x, t) is an unknown for (x, t) =...
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| Main Authors: | , |
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| Format: | Journal |
| Published: |
2018
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| Subjects: | |
| Online Access: | https://www.scopus.com/inward/record.uri?partnerID=HzOxMe3b&scp=78649839395&origin=inward http://cmuir.cmu.ac.th/jspui/handle/6653943832/49228 |
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| Institution: | Chiang Mai University |
| Summary: | In this paper, we study the nonlinear equation of the form ∂/∂t u(s, t) + c2(-*)ku(x, t) = f(x, t, u(x, t)), where*k is the operator iterated k-times, defined by*k=[(Σi=1p ∂2/ ∂xi2)3 + (Σj=p+1 ∂2/∂xi2)3]k, where p + q = n is the dimension of the Euclidean space ℝn, u(x, t) is an unknown for (x, t) = (x1,x2. . . , X n, t) ∈ ℝn × (0, ∞), k is a positive integer and c is a positive constant, f is the given function in nonlinear form depending on x, t and u(x, t). On suitable conditions for f, p, q, k and the spectrum, we obtain the unique solution u(x, t) of such equation. Moreover, if we put q = 0, k = 1, we obtain the solution of non-linear heat equation. © 2009 Academic Publications. |
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