J. Nonl. Mod. Anal., 6 (2024), pp. 841-872.

Published online: 2024-08

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Consider the Cauchy problem for the Benjamin-Ono-Burgers equation. There exists a unique global weak solution under appropriate conditions
on the initial function and the external force. Here are many very important
and interesting questions.

• Can we accomplish the exact limits for all order derivatives of the global
smooth solution of the Benjamin-Ono-Burgers equation, in terms of some given
information, representing certain physical mechanisms?

• What are the influences of various physical mechanisms (represented by the
initial function, the external force, the order of the derivatives and the diffusion coefficient) on the exact limits?

• Can we establish improved decay estimates with sharp rates for all order
derivatives of the solution, so that the most important constants $\mathcal{A}$ and $\mathcal{C}$ are
independent of any norm of any order derivatives of the initial function, the
external force and the solution, for all sufficiently large $t > 0?$ Other positive
constants $\mathcal{B}$ and $\mathcal{D}$ in the estimates are much less important because $Bt^{−1}$ and $Dt^{−1}$ becomes arbitrarily small as $t → ∞.$ This kind of decay estimates may
play a substantial role in long time, accurate numerical simulations.

• Can we use the solution of the corresponding linear equation to approximate
the solution of the Benjamin-Ono-Burgers equation?

• Can we couple together classical ideas (such as the Fourier transformation,
the Parseval’s identity, Lebesgue’s dominated convergence theorem, squeeze
theorem, etc) in an appropriate way to establish important and interesting
results for the Benjamin-Ono-Burgers equation?

• For very similar dissipative dispersive wave equations, such as the Kortewegde Vries-Burgers equation and the Benjamin-Bona-Mahony-Burgers equation,
can we apply the same ideas developed in this paper to establish the same or
very similar results?

We will couple together a few novel ideas, several existing ideas and existing
results and use rigorous mathematical analysis to provide positive solutions to
these important and interesting questions.

Consider the Cauchy problem for the Benjamin-Ono-Burgers equation. There exists a unique global weak solution under appropriate conditions
on the initial function and the external force. Here are many very important
and interesting questions.

• Can we accomplish the exact limits for all order derivatives of the global
smooth solution of the Benjamin-Ono-Burgers equation, in terms of some given
information, representing certain physical mechanisms?

• What are the influences of various physical mechanisms (represented by the
initial function, the external force, the order of the derivatives and the diffusion coefficient) on the exact limits?

• Can we establish improved decay estimates with sharp rates for all order
derivatives of the solution, so that the most important constants $\mathcal{A}$ and $\mathcal{C}$ are
independent of any norm of any order derivatives of the initial function, the
external force and the solution, for all sufficiently large $t > 0?$ Other positive
constants $\mathcal{B}$ and $\mathcal{D}$ in the estimates are much less important because $Bt^{−1}$ and $Dt^{−1}$ becomes arbitrarily small as $t → ∞.$ This kind of decay estimates may
play a substantial role in long time, accurate numerical simulations.

• Can we use the solution of the corresponding linear equation to approximate
the solution of the Benjamin-Ono-Burgers equation?

• Can we couple together classical ideas (such as the Fourier transformation,
the Parseval’s identity, Lebesgue’s dominated convergence theorem, squeeze
theorem, etc) in an appropriate way to establish important and interesting
results for the Benjamin-Ono-Burgers equation?

• For very similar dissipative dispersive wave equations, such as the Kortewegde Vries-Burgers equation and the Benjamin-Bona-Mahony-Burgers equation,
can we apply the same ideas developed in this paper to establish the same or
very similar results?

We will couple together a few novel ideas, several existing ideas and existing
results and use rigorous mathematical analysis to provide positive solutions to
these important and interesting questions.

*Journal of Nonlinear Modeling and Analysis*.

*6*(3). 841-872. doi:10.12150/jnma.2024.841