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Nasser
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use FrameLabel since you set Frame to True. You were using FrameLabel And also using PlotLabel. Why mix the two? FrameLabel can do all of these. FrameLabel is automatically centered on Frame.

Mathematica graphics

Plot[Sin[t], {t, 0, 50},
 PlotStyle -> Red,
 Axes -> False,
 Frame -> True,
 PlotTheme -> "Classic",
 FrameStyle -> Directive[Black, 25],
 PlotLegends -> Placed[{"Exact."}, {{1, 0.56}}],
 FrameLabel -> {{"ϕ (0, t)", Automatic}, {"t", 
    Style[Framed["w = 1, k = 0", RoundingRadius -> 5], 20, Black, 
     Background -> Lighter[White]]}},
 ImageSize -> 500
 ]
Nasser
  • 150.5k
  • 12
  • 161
  • 375